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METHUEN'S MONOGRAPHS 




ON PHYSICAL SUBJECTS 




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X-RAYS 


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.nattbuen'g jCBonograpftg on ffftpgital ^ufajtctg 
General Editor: B. L. Worsnop, B.Sc, Ph.D. 



X-RAYS 



METHUEN'S MONOGRAPHS ON PHYSICAL 
SUBJECTS 

F'cap Svo, 2a. 6d. net each 

General Editor: B. L. Woesnop, B.Sc, Ph.D. 
Lecturer in Physios, King's College, London. 

SPECTRA. By R. 0. Johnson, D.Sc, Lecturer in 
Physics, King's College, London. 

WAVE MECHANICS. By H. T. Flint, Ph.D., D Sc 
Reader in Phyaics in the University of London! 
3«. od. net. 

THE PHYSICAL PRINCIPLES OF WIRELESS By 
J. A. Ratcliffe, M.A., Sidney Sussex College, Cam- 
bridge. 

^CONDUCTION 0F ELECTRICITY THROUGH 
GASES. By K. G. Emeletjs, M.A., Ph.D., Lecturer 
in Physics, Queen's University, Belfast. 

MAGNETISM. By E. C. Stoner, Ph.D., Reader in 
.Physics at the University of Leeds. 

X-RAYS. By B. L. Worsnop, B.Sc, Ph.D. 

X ^ Y CRYSTALLOGRAPHY. B y R - W. James, 
M.A., B.Sc, Lecturer in Phyaics in the University of 
Manchester. * 

^ COMMUTATOR MOTOR. By F. J. Teago, D.Sc , 
M.1.JS..I!.., Professor of Electrical Machinery in the 
University of Liverpool. 

In Preparation 

THE APPLICATIONS OF INTERFEROMETRY. By 
W. Ewaet Wiixiams, M.Sc, Lecturer in Physics, 
King's College, London. ' 

T 2 E a T J?f RM * 0NIC VALVE. By E. V. Appleton, 
M.A., D.Sc. F.R.S., Professor of Physics in the Uni- 
versity of London. 

TA , BL J S T / PH YSICAL CONSTANTS. By Dr 
W. H. J. Childs. 

PHOTOCHEMISTRY. By D. W. G. Style, B.Sc, Ph.D. 



X-RAYS 



BY 

B. L. WORSNOP, B.Sc, Ph.D. 

LECTURER IN PHYSICS, KING'S COLLEGE, LONDON 

WITH A PREFACE BY 

O. W. RICHARDSON, F.R.S. 

YARROW RESEARCH PROFESSOR OP THE ROYAL SOCIETY 
NOBEL LAUREATE IN PHYSICS, 1928 



WITH 36 DIAGRAM9 




METHUEN & GO. LTD. 

36 ESSEX STREET W.C. 

LONDON 



First Published in IQ30 






GENERAL PREFACE 

THIS series of small monographs is one which should 
commend itself to a wide field of readers. 

The reader will find in these volumes an up-to-date 
resume of the developments in the subjects considered. 
The references to the standard works and to recent 
papers will enable him to pursue further those subjects 
which he finds of especial interest. The monographs 
should therefore be of great service to physics students 
who have examinations to consider, to those who are 
engaged in research in other branches of physics and 
allied sciences, and to the large number of science 
masters and others interested in the development of 
physical science who are no longer in close contact with 
recent work. 

From a consideration of the list of authors it is clear 
that the reader need have no doubt of the accuracy of 
the general accounts found in these volumes. 

0. W. RICHARDSON 

King's College 
London 



PRINTED IN GREAT BRITAIN 



PREFACE 

THIS book was written, in the first instance, as an 
answer to several inquiries about the properties 
of X-rays which were addressed to the writer by research 
workers in other branches of physics and by advanced 
students. These inquiries revealed a need, not only in 
X-rays but in most branches of physical and allied 
sciences, for concise accounts of the principles and 
present position of each subject, and this series of 
monographs was undertaken in an attempt to meet 
what was felt to be a real demand. j 

In the following pages is set out an ' outline of X-rays. 
The early work is summarized ; some detail is given 
of the determination of wave-length ; the remaining 
chapters deal in turn with the more important physical 
properties of the rays ; and the work concludes with a 
final chapter in which an account is given of the recent 
methods of investigation of diffraction, reflection and 

refraction. . , 

The book is not addressed to the specialist, but rather 
to those who have a scientific training and a wish to 
have a statement of the present position m X-rays, 
either from a desire for general information, or as an 
introduction to a detailed reading of the subject^or as 
a preparation for the examination for B.Sc. From a 
long experience of those medical practitioners who 
are reading for the Cambridge Diploma in Radiology 
the writer feels sure that a general perusal of the book 
(neglecting the ' mathematics ') will be of some use to 

The diagrams have been specially drawn for the book, 
but the writer wishes to acknowledge that many of them 

vii 



Vlll 



X-RAYS 



are inspired by the standard works and original papers 
on the subject. 

It is a pleasure to record the fact that Dr. L. Simons 
read the proof sheets and made suggestions, some of 
which have been incorporated in the text. 

a, B. L. WORSNOP 

Wheatstone Laboratory 
King's College 

London 
January, 1930 






CONTENTS 



CHAP 

I 



n 



in 



IV 



VI 






Introductory and Early Experiments 

Introduction. Intensity. Absorption coefficient. 
Secondary radiations. Polarization. Polariza- 
tion of secondary radiation. The wave nature of 
X-rays. 

The Determination of Wave-length and X-ray 
Spectroscopy . 
Reflection at crystals. The X-ray spectrometer. 
Crystal structure. Determination of d and A. 
Moseley's law. Bohr's theory. Development of 
technique. Limitations to the use of crystals. 

The Scattering of X-rays . • 

Thomson's theory of scattering. Distribution of 
the scattered rays. The scattering coefficient. 
Wave-length of the scattered rays. Compton s 
theory of scattering. 

Emission and Absorption . 

Emission lines. The K series. The L series. 
The M and N serios. General conditions of excita- 
tion. General radiation. The total intensity of 
the general radiation. Absorption. Photographic 
methods. Absorption and wave-length. Atomic 
nature of absorption. 

Photo -electrons and Ionization 

Introductory. Absorption and range. Experi- 
mental methods. Experimental results. 

Direct Reflection, Refraction and Diffraction 
Early experiments. Deviation from Bragg's law. 
Reflection. Refraction by prisms. The Reflec- 
tion grating. 

Bibliography 

Index 



IX 



PAGE 
1 



19 



40 



53 



72 



88 



07 
99 



X-RAYS 

CHAPTER I 

INTRODUCTORY AND EARLY 
EXPERIMENTS 

Introduction 

WHEN a sufficiently large potential difference is 
applied to two electrodes in an exhausted 
vessel, cathode rays are given off normally to the 
cathode. Inquiry into the nature of the cathode rays 
led to many experiments and much speculation towards 
the end of last century. 1 Crookes suggested a corpus- 
cular nature for the ' rays ', whereas the majority of 
the continental workers thought them to be a radiation. 
Roentgen, working on the problem in 1895, found 
' if the discharge from a fairly large induction coil be 
made to pass through a Crookes tube or a Hittorf 
tube which had been sufficiently exhausted' that 
fluorescence was produced in crystals of barium platino- 
cyanide placed at distances of up to 2 metres away 
from the tube, which itself was entirely covered with 
black paper. This he traced to the effect of a radia- 
tion, quite distinct from the cathode rays, which had 
an origin at the place where the cathode rays were 
stopped. Referring to this, he says: 'for brevity's 
sake I shall use the expression " rays ", and to dis- 
tinguish them from others of this name I shall call 
them " X-rays ".' 

1 See Emeleus" Conduction of Electricity through Geuw. 

I 



2 X-RAYS 

In addition to the fluorescence produced in these 
crystals, X-rays were found by Roentgen to have the 
property of affecting a photographic plate and of 
producing ionization in gases, i.e. making gases con- 
ductors of electricity. No success attended the experi- 
ments, which were designed to show reflection, refraction 
and interference which were to be expected if X-rays 
were a true radiation. 

It was found that some substances which were 
opaque to light were transparent to X-rays, and 
generally it appeared that with X-rays the opacity of 
substances was dependent on their atomic weight and 
density, e.g. bone was more absorbing than tissue- 
lead cut off the rays more than an equal thickness of 
wood or paper. An early application of this was 
found in medicine, where shadow pictures of the bones 
of the hand, etc., proved at least one use which the 
new rays might have. This and the normal physicist's 
development led to the construction of special X-rav 
tubes. J 

The Crookes tube was replaced by the focus tube 
as illustrated in Fig. 1, in which a concave cathode 
brought the cathode stream to a focus at the focal 
spot on a heavy anticathode. To assist the running 
of the tube, especially in the suppression of the inverse 
current when an induction coil was used, an anode was 
placed m a side tube, and was joined to the anticathode 
as shown in Fig. 1. 

The penetrating power of the rays depends on the 
potential required to generate them. This in turn is 
dependent on the residual gas pressure in the tube. 

* u ° e m wfa ich the pressure is relatively low requires 
a high potential to pass a current through it, and the 
tube and the rays are said to be ' hard ' ; when the 
gas pressure is higher, the tube is ' soft ' and the rays 
less penetrating. In all tubes of this type the current 
is passed by means of ionized gas molecules. When 
tjie positive ions strike the cathode, cathode rays or 
electrons are ejected and move down the tube with a 






EARLY EXPERIMENTS 3 

velocity appropriate to the applied potential. When the 
cathode rays are stopped their energy is mostly converted 
into heat, and but a small fraction appears as radiation. 
The excessive heat developed at the anticathode necessi- 
tates the use of a metal of high melting-point : m many 
tubes the anticathode is hollow and a continuous 
stream of water prevents undue heating of this part. 

In this connexion also it is to be noticed that most 
metals sputter 1 when used as cathode in such a tube. 
The result is that the walls of the tube become coated 
with a thin layer of the metal. From this point of 



Softer":' - Mico Shetf 




Fig. 1 

view aluminium is the best substance to use as cathode, 
as it produces a relatively small amount of sputtering. 

To vary the ' hardness ' of the type of tube described 
above, a device seen in Fig. 1 has been introduced. A 
side tube which contains mica or certain of the rare 
earths may be made to take part of the current by 
connecting to the anode and cathode for a very short 
interval. This discharge through the softener liberates 
sufficient occluded gas to soften the tube. The process 
of hardening may be carried out by a long small dis- 
charge through the tube, and is in fact done in normal 
running. 

1 See Emeleus' Conduction of Electricity through Gases. In 
this series (p. 59). 






4 X-RAYS 

The tube just described is called a gas tube and was 
tne type with which most of the early work on X-rays 
was accomplished. The design remained almost un- 
altered until about 1912, when Coolidge in America 
and idhenfeld m Germany introduced the idea of a 
cathode stream generated independently of the potential 
winch accelerates the electrons. 

Fig. 2 shows the essentials of the Coolidge tube 
which is the better known in this country. The tube 
is exhausted to such an extent that no current will 
pass through it even when very high potentials are 
applied. Ihis is because at reduced pressure a charged 
ion may pass from cathode to anode without colliding 



Heaty TVngsfen 
Ann ca mode 




Fia. 2 

with an atom ; i.e. the ' mean free path ' of the ion 
is greater than the effective length of the tube The 
current carried by isolated ions is negligibly small ■ 
it requires ionization by collision to produce a current 
measurable m milliamperes as obtains in a gas tube 
Under these circumstances no cathode stream is spon- 
taneously developed. The cathode is a flat spiral of 
tungsten wire connected to an external source of 
potential of some 14 volts obtained from accumulators 
or a transformer. The current through the cathode 
nlament is so regulated that the thermionic emission 
ot electrons is sufficient to produce the required amount 
oi ^-rays, when the high potential from a transformer 
or induction coil is applied to give the requisite velocity. 
ihe electron stream is ' focussed ' on the anticathode 






EARLY EXPERIMENTS 5 

by means of the electrostatic repulsion from the 
molybdenum cap which surrounds the cathode and 
which is joined to it. The characteristic feature of 
the Coolidge tube is therefore its adjustability. The 
quantity of X-rays may be varied by the filament 
current, and the penetration may be varied by the 
applied potential. Also the tube may be set at any 
condition of hardness or intensity at anytime, and those 
conditions may be repeated at will, whereas with a gas 
tube this is not so. 

Intensity 

The intensity of a beam of X-rays is defined as the 
energy passing per second through unit area placed 
normal to the beam at the point under consideration. 

It has been shown that the intensity follows the 
inverse square law, i.e. it falls off inversely as the square 
of the distance from the source. It may be measured 
by the blackness of the photographic image it produces 
in a given time ; by the intensity of the fluorescence in 
powdered crystals, or by the magnitude of the ioniza- 
tion set up in a gas. 

The ionization produced by an X-ray beam sets up 
a current, between two points at different potentials, 
which is proportional to the intensity of the beam 
provided that all the rays are equally efficient in the 
production of ions. This condition holds for X-rays of 
similar hardness, and the intensity of such beams 
therefore may be compared by ionization methods. 

Fig. 3 shows the essentials of an experimental arrange- 
ment for this purpose. The beam is limited to a narrow 
pencil by lead screens, S, S, and passes through a thin 
aluminium window in an aluminium ionization chamber, 
I. A collecting plate or rod, P, is connected through 
an insulating plug to a quadrant electrometer, Q.E, 
and is arranged so that the beam cannot impinge 
directly upon it. The ionization chamber, which is 
lined with several sheets of paper, is raised to a potential 



6 



X-RAYS 



sufficiently high to drive the ions of one sign to the 
collecting rod without appreciable recombination, i.e. 
the space is saturated. 

The passage of the X-rays through the ionization 
chamber, therefore, causes the quadrant electrometer 
to charge up. For a fixed potential on the ionization 
chamber the charge received per second by the collect- 







ing system depends upon the number of ions produced 
per second in the space, that is upon the intensity of 
the X-ray beam. Since the capacity of the system is 
constant, the amount of electricity arriving per second (or 
in other words the current measured) is proportional 
to the deflection per second. The rate of movement 
of the needle of the quadrant electrometer is therefore 
proportional to the intensity of the beam entering the 
chamber. 



EARLY EXPERIMENTS 
Absorption Coefficients 



If a sheet of aluminium is placed at A, it is found 
that the intensity is reduced. When the beam is 
homogeneous, i.e. not made up of a heterogeneous 
mixture of rays of different hardness, it is found that 
successive equal thicknesses of aluminium reduce the 
intensity by equal fractions. 1 In these ideal conditions 
we refer to the linear absorption coefficient of the 
homogeneous rays, as being the fractional reduction in 
the intensity of the beam by unit path in the absorbmg 

substance. . 

An alternative definition of the absorption coefficient 
may be deduced, for, if the reduction in the intensity 
of the beam is dl when it traverses a thickness dx of 
the absorber, we have from above — dl/1 cc dx, or, 
— dl/1 = fxdx, where \i is a constant (the absorption 
coefficient of the rays in the material of the absorber) 
and I is the initial intensity of the beam. 

Integrating, we have log. I/To = — P x 

or I = I e -"* (1) 

where I and I are the intensity before and after pass- 
ing through a thickness, x, of the absorber. 

In an experimental determination the ratio of the 
ionization currents as measured by the quadrant 
electrometer replaces the ratio I/I 0) and so the value of 
[i may be obtained by substitution in the following 
formula, where d and d are the rates of movement 
of the needle of the quadrant electrometer : 
2-3 x log, d /d 
**- 1 • 

An alternative arrangement of apparatus for this 
determination is seen in Fig. 3(6). In this case a gold- 
leaf electroscope is used : the plate, P, and the leaf are 

1 E.g. if one thickness of absorber reduces the incident in- 
tensity, J, to 9/10 /, and the transmitted beam falls on a 
second equal absorber, the intensity transmitted by this sheet is 
9/10 (9/10 J) — -81/ and so on. 
2 



8 



X-RAYS 



raised to a high potential. The measurement made is 
the rate of discharge of the electroscope over a fixed 
part of the scale in the eye-piece of the observing low- 
power microscope. A similar calculation to that above 
enables one to find ft. But in this method, as described, 
as in the preceding one, the value of /• obtained is only 
reliable if special precautions have been taken to ensure 
a steady output from the X-ray tube. If, as is usual, this 
latter is liable to fluctuate, compensation must be made. 
One typical method is to use two ionization chambers 
and electroscopes : the one system is placed directly 
in a narrow beam of X-rays from the tube and is used 
as a standard ; the other is employed in much the 
same way as described above. The method of working 
is to find the number of divisions (in the eye-piece scale) 
traversed by the one gold leaf, while the gold leaf of 
the electroscope connected to the standard chamber 
moves over a certain range. 

This is repeated for each thickness of absorber 
employed in the measurement, and in this way, even 
if the tube fluctuates the movement measured corre- 
sponds to a definite amount of X-ray energy passing 
through the measuring system. The function of the 
standard ionization chamber is to take the place of 
the clock in the simpler method. 1 

Other devices have been used to overcome tube and 
coil variations, viz. the compensated steady deflection 
method, 2 in which the ionization current is passed to 
earth through a very high resistance and the steady 
potential at the ends of the resistance is measured by a 
quadrant electrometer. The above however shows the 
general principles which underlie the methods. 

The absorption coefficient as denned above is also 
the fractional reduction in the intensity of the homo- 
geneous beam per unit volume of absorber as may be 

1 Air is used as the gas in the standard chambor, as otherwise 
a change in quality of the incident rays might produce anomalous 
results. 

* For example, see p. 38. 



EARLY EXPERDfENTS 



B 






readily seen if one remembers that the intensity is 
defined in terms of energy per unit area of cross section 
of that beam. For a fixed source of X-rays fi is 
constant only so long as the physical state and condi- 
tion of the absorber remains unchanged. For a gas fi 
increases with increase of pressure ; the solid state of 
the absorber has a greater fi than the gaseous state. 
If we calculate the ratio /x/p, however, where p is the 
density of the absorber we obtain a quantity which is 
independent of the physical state and conditions. 
This quantity is called the mass absorption coefficient 
of the substance as it is the fractional reduction per unit 
mass, the mass being supposed to be in the form of a 
slab 1 sq. cm. in area. 

The atomic absorption coefficient, or the fraction 
absorbed per atom, /i a , is the quotient [i/N, where N 
is the number of atoms per c.c. This, like the mass 
absorption coefficient, is independent of the physical 
state of the substance, except, it may be, for a very 
slight linear increase with increase in temperature. 

The radiation which is given out by the anticathode 
of an X-ray tube is by no means homogeneous, but it is 
often convenient to refer to the average penetrability 
of the beam as giving an approximate estimate of the 
1 hardness ' of the rays. The value of /.i/p quoted in 
this case is that calculated from the thickness of absorber 
required to reduce the intensity of the rays to half value. 

In the early work the only test for homogeneity was 
by absorption. The so-called ' homogeneous ' rays 
were those which gave a linear relation between log. I 
and x. Analysis of this kind led Barlda to the conclu- 
sion that the secondary K and L radiations which he 
discovered were homogeneous, as shown in the next 
section. 

Secondary Radiations 

When X-rays fall on matter, some of the beam passes 
through with a reduced intensity as already described, 
but in addition to this the matter becomes the source of 



10 



X-RAYS 



the so-called secondary radiations. These were initially 
separated out by analysis of absorption curves (curves 
showing the relation between log. I and a) and are : 

(1) Scattered radiation, which in almost every 
respect is identical with the incident beam, and may 
be compared to light scattered by tobacco smoke. It 
passes out from the scattering substance in all direc- 
tions to a greater or less extent. 

(2) Electronic radiation, which consists of electrons 
ejected from matter by the incident beam and is there- 
fore not a true radiation, but is like the photo-electric 
emission set up when ultra-violet light falls on sub- 
stances. 

(3) Homogeneous radiations, which are character- 
istic of the substance irradiated. They are always of a 
less penetrating nature than the incident beam. Barkla 
found, from absorption measurements, that the range of 
elements from Ca to Ag gave out a single homogeneous or 
characteristic radiation, while those elements of atomic 
weight greater than that of silver gave two such radia- 
tions ; e.g. when barium was irradiated by a penetrating 
beam, and precautions were taken to eliminate the 
scattered and electronic radiations, it was found that 
the remaining beam had two components, one soft, one 
hard. This was the general case with the heavy 
elements. When the penetration of these two com- 
ponents was obtained in terms of the fz/p in aluminium, 
it was found that the values for the different elements 
could be arranged in two series as is shown in Fig. 4. 
Barkla called the more penetrating rays (i.e. the rays 
with the smaller ii/p) the K series, and the lesser pene- 
trating (large Lt/p) the L series. 

As stated above, it was found that K rays only were 
obtained from the light elements, the presumption 
being that the L rays of these elements were so soft 
that they were absorbed in the elements themselves 
and in the surrounding air. Characteristic radiation 
may be produced only when the incident radiation is 
more penetrating than that to be excited, i.e. it is a 



EARLY EXPERIMENTS 



11 



case in which Stokes's law always holds. This type of 
secondary radiation is found to be equally distributed 
in all directions about the source, 

With modern technique, using vacuum spectro- 
meters, it has been possible to isolate the K series of 
the lighter elements and the range of this series has 
been so extended to sodium. In addition it has been 
found that the heavier elements give out characteristic 




30 ioo '30 too &oC no 

Aromtc tYcigt>f of Mefcl emitting Radiation. 

Fio. 4 



rays of a less penetrating nature than the L, which 
Siegbahn has called the M series, and further, in the 
case of uranium, thorium, and bismuth a softer N series 
has been established. 

In no case has a more penetrating radiation of a J 
series been established, although there was evidence 
which pointed this way at one time (page 46). 

An interesting insight to the condition of excitation 
of the K and L series is to be had from a consideration 



12 



X-RAYS 



of Fig. 5, which shows the absorption coefficient of 
different rays as measured in one absorber, say, nickel. 
Passing from right to left, corresponding to a decrease in 
wave-length of the incident radiation, the absorption 
coefficient of these rays as measured in nickel decreases 
until a point L is reached. The incident beam is here 
sufficiently hard to excite the L radiation of nickel, 
and there is a corresponding increase in fi/p ; the antici- 
pated decrease of this quantity then proceeds until K 
is reached, when the incident beam is hard enough to 




Hora 4 fcys Qua/ity of Xftay (ty In A/) 

— - /ncre"*>ng IVorc/enc/m 

Fig. 5 



J 'off Pay I 



excite the K rays of nickel. Associated with this is 
the extra absorption seen in the figure. Subsequently 
the ordinary decrease in absorption in nickel as the 
penetration of the incident beam increases, is seen to 
occur. 

The process taking place in the absorption of energy 
which is obviously associated with the emission of 
characteristic rays, is that electrons are dragged from 
their stable position within the atom and may appear 
as part of the electronic radiation. The falling in of 
electrons to take up the vacant positions so created, 
results in energy being radiated as homogeneous radia- 
tion. This is further discussed later in Chapter II. 

It was shown by Kaye and others that the beam from 



EARLY EXPERIMENTS 13 

an X-ray tube possessed, in addition to the 'general 
radiation ' (i.e. the heterogeneous rays), a strong com- 
ponent of rays characteristic of the material of the 
anticathode of the tube, provided that the applied 
potential was sufficiently great. It appears therefore 
that characteristic rays may be excited by electrons 
as well as by other X-rays. This method is now the 
usual one used to produce characteristic radiations. 
See, for example, Chapter II. 

Polarization 
A somewhat picturesque way of regarding the genesis 
of general X-rays by the stoppage of a moving electron, 
may be had by considering the electron as a charge 
moving with its lines of force spreading out in all direc- 
tions. When a sudden impact arrests its flight, an 
equally sharp pulse is transmitted along the lines of 
force, constituting the X-ray. This simple picture was 
the basis of the early theories of Stokes and J. J. 
Thomson but finds little support in the modern theory. 
Nevertheless it gives a mental picture which is useful 
to anticipate just those properties of X-rays which are 
still best explicable in terms of the older classical 

1 f 1 Ofl R 

According to classical mechanics an electron radiates 
whenever its velocity is changed. If we consider a 
parent cathode ray stream OA striking an anticathode 
at A, Fig. 6, the velocity of the electrons is changed in 
the direction OA and therefore the electric vector of the 
radiation so produced is in a direction parallel to OA, 
and, in particular, the electric vibrations in the rays 
which proceed in the direction AB are in the plane of 
the paper ; no vibrations will occur in a direction at 
right angles to this plane. In other words the X-rays 
are plane polarized. This may be visualized by using 
the line of force picture. 

Such complete polarization is not to be expected in 
practice however, as the electron is not stopped at one 
impact, but has a further life in a deflected direction 



14 



X-RAYS 



before it is finally brought to rest, and it therefore gives 
rise to additional radiation with vibrations in all possible 
directions. It is to be expected, however, from the 
considerations, that the beam is at least partially 
polarized. 

This theory was first put to the test by Barkla in 
1906 ; he found the intensity of scattered radiation 
set up in two directions such as BD, in the plane of the 
diagram, and BC at right angles to this plane. His 
measurement showed that there was a maximum 



Max perpoc/icu/o' 
to plane of paper. 




Fig. 6 

amount of ionization in an electroscope at C and a 
minimum at D. If we think of the electrons in the 
scattering substance as being set in motion by the 
electric vector in the incident beam, it is clear from the 
diagram that such an up and down movement of the 
electrons at B will result in radiation in the plane normal 
to the diagram and in particular along the direction 
BC, but a radiation of this sort is not to be expected 
in the direction BD. Barkla found the difference in 
the two directions to correspond to a partial polariza- 
tion of some 10 to 20 per cent. 



EARLY EXPERIMENTS 



15 









This was confirmed with elaborations by Bassler and 
others, and it was found that the polarization was more 
complete if the beam was first filtered to remove the 
softer components, as is to be expected if the softer 
rays have their origin in the subsequent arrest of the 
deflected cathode rays as already described. It was 
also found that an increase in the velocity of the parent 
cathode ray stream reduced the amount of polarization, 
presumably due to these electrons being stopped by a 
number of impacts. 

POLARIZATION OF THE SECONDARY RAYS 

Barkla investigated the polarization of the scattered 
beam in a way very similar to that just considered. A 
primary beam took the place of the cathode stream OA, 
a radiator gave rise to a scattered beam AB, and two 
electroscopes tested the relative intensities of the 
tertiary rays scattered at a carbon block at B in the 
two directions at right angles. To perform this experi- 
ment the parent cathode ray beam was at right angles 
to the figure, at 0, and consequently the electroscope D 
showed a maximum and C a minimum. The difficulty 
in these experiments was that the intensity of the 
tertiary rays was very small, and consequently wide 
apertures had to be used, with the result that the 
electroscopes were not receiving rays which had all 
been scattered at 90°, also, to obtain appreciable intensity 
the scattering sheets were thick and so the effects of 
the interaction of the scattered rays in the different 
depths was a disturbing factor. In spite of these 
disadvantages, which were inherent in the experiments 
of that time, Barkla was able to show that the scattered 
beam was at least 75 per cent, polarized. Recent 
experiments by Compton and Hazelnow, using intense 
beams, thin radiators and small apertures, have shown 
that, within experimental error, the polarization is 
complete, as the classical theory predicts. 

Investigation of the homogeneous radiation was 



lfi 



X-RAYS 



EARLY EXPERIMENTS 



17 



also made ; it was shown that in this case there was no 
polarization. 

The Wave Natuee of X-rays 

When announcing his discovery of X-rays Roentgen 
suggested that they might be longitudinal waves, but 
his endeavours to produce reflection, refraction and 
interference were unavailing. 

Polarization and scattering experiments, however, led 
to the conclusion that X-rays were transverse waves. 
Haga and Wind, and later Water and Pohl, endeavoured 
to measure the wave-length by diffraction at fine wedge- 
shaped slits. They estimated the order of the wave- 
length at not less than 10" 9 cm., but, due to the con- 
gested nature of their diffraction pattern, their results 
were not very convincing without further independent 
experiments. 

Magnetic deflection experiments, described later, en- 
abled a calculation of the velocity of the corpuscular 
radiation excited by X-rays to be made and, by an 
application of the quantum theory, the frequency of 
the X-rays used in their ejection was calculated. 

From the above and other sources the probable order 
of the wave-length was 10" 8 to 10" 9 cm., i.e. of the same 
order as atomic dimensions. This led Laue in 1912 to 
make the most profitable suggestion that in the regular 
arrangement of atoms in a crystal there was a suitable 
means of producing diffraction of X-rays, in a way 
similar to that provided for visible radiation by a line 
grating. 

The idea was that if a narrow pencil of X-rays be 
passed through a crystal, the individual atoms should 
set up secondary wavelets which, according to Laue, 
should reinforce in certain directions, in addition to the 
directly transmitted direction. At his suggestion Fried- 
rich and Knipping undertook experiments on these lines 
to test the theory. A narrow pencil of X-rays was 
limited in the usual way by lead stops and directed 
normally on to a thin slip of crystal and was received 






i 



on a photographic plate placed some 15 cm. away on 
the side of emergence. The photographic plate showed 
a central spot surrounded by a symmetrical system of 
spots, as seen in Fig. 7. When the plate was moved 
further away the size of the spots remained the same, 
but the pattern spread out and was of the same form ; 
rotation of the crystal resulted in a rotation of the 
pattern, which was therefore seen to be produced by 
the crystal and which was the experimental realization 
of the results which Laue had anticipated by a mathe- 
matical analysis. 

The accepted idea of 
the structure of crystals 
was some regular arrange- 
ment of atoms. This was 
the basis of the develop- 
ment referred to above. 
The constant form of 
crystals had led to the 
belief that a crystal was 
made up by a repetition 
of a fundamental unit. 
For example, if we con- 
sider the simplest case 
of the cubic crystal, 

which, like common salt, is in the form of cubes 
no matter how small be the crystal, the fundamental 
unit to assume is that of a cube. The unit may be 
considered to be made up of eight atoms set at the 
corners of this cube, in which case it is called a simple 
cubic system ; or this may be modified by having an 
additional atom at the centre of each face, when it is 
said to be a face centred system ; or again a modification 
is to be had if the unit has an additional atom at the 
centre of the cube, and is called a cube centred system 
(see Fig. 14). 

Repetition of any of these units results in a regular 
spacing of atoms within the crystal which constitutes 
the space lattice. 




Fia. 7 



18 



X-RAYS 



On this supposition there will occur planes rich in 
atoms parallel to the faces of the crystal and equally 
spaced throughout. For example, in Fig. 8, which 
shows the crystal lattice in two dimensions only, it may 
be seen that vertical columns and horizontal rows satisfy 
this condition. W. L. Bragg suggested that there will 
occur other planes less rich in atoms which would yet 
serve as reflecting planes for the X-rays (in Fig. 8 the 
planes, represented by thin lines, are examples of this 




Fro. 8 

kind of thing). It was shown that these planes would 
be symmetrical with respect to the incident beam, and 
that the spots of the Laue patterns could be accounted 
for in terms of ' reflection ' at these planes throughout 
the crystal. This gives a much simpler way of regard- 
ing the Laue pattern than the complex analysis originally 
used. 

W. H. and W. L. Bragg then tried, successfully, to 
bring about ' reflection * at the planes parallel to the 
cleavage surfaces of crystals. The success of these ex- 
periments, using an ionization method, led them to con- 
struct what is now known as the X-ray spectrometer by 
means of which the determination of wave-length and 
an accurate knowledge of crystal structure has been 
deduced. 






CHAPTER II 

DETERMINATION OF WAVE-LENGTH 
AND X-RAY SPECTROSCOPY 

Reflection at Crystals 

IF a beam X-ray were to fall on to a single layer of 
atoms such as, say, one plane of a crystal containing 
a regularly spaced set of atoms, we should expect a very 
weak reflection for all angles of incidence and for all wave- 
lengths, and the problem of X-ray reflection would be 
analogous to the reflection of light at a polished surface. 

In dealing with X-rays, however, the high polish of 
the surface does not bring about reflection as in the case 
of light. X-rays penetrate the very large number of 
atom layers contained in the thin surface sheath of the 
crystal, and each of the atoms irradiated sets up a new 
wave with an intensity which depends on the number 
of electrons within the atom. Therefore, in order to 
determine if there is any reflected ray corresponding to 
a ray incident at a given angle, we must find the integral 
effect of the secondary wavelets, as is done in the simpler 
case of Huyghcns' construction in optics. The problem, 
in other words, is one of reflection at a very large 
number of parallel reflecting surfaces. 

Suppose that in Fig. 9, L 1( L 2 , L 3 represent the layers 
of atoms in a crystal, separated by a distance d, called 
the lattice constant, which is of the order of 10" 8 cm. If 
the secondary wavelets from the adjacent layers are to 
reinforce each other in a given direction, they must be 
exactly in the same phase in that direction (i.e. the crests 
of the waves must fit together and not a crest and a 

19 



20 



X-RAYS 



trough together). The slightest departure from this 
condition results in interference of the rays with a 
reduction in intensity to very minute values. 

Let us consider a narrow beam of rays incident at a 
glancing angle 6. In Fig. 9 five of the adjacent rays of 
this beam are shown. They are in phase at R, S, T, U, V. 
The ray R strikes an atom in the first layer at A and 
sends out a secondary wave : the ray S strikes an 
atom of the second layer at and so on, and in the 
direction QM, which is also inclined to the crystal at 6°, 
there is a contribution from all the secondary wavelets. 
Now ray 2 has a longer path than ray 1 ; ray 3 has 




an equal excess path over ray 2, and so on. To reach 
A the rays traverse the paths RA, SOA, TPA, etc. 
Now SOA = SO -f- OA, and the geometry of the figure 
shows that OA = OB, also that RA is equal to SN 
where N is the foot of the perpendicular from A to SO, 
therefore the extra path traversed by ray 2 is equal to 
NB, which is clearly equal to 2 x d sin 0, which there- 
fore represents the extra path traversed by successive 
neighbouring rays. If this extra path is equal to one 
whole wave-length of the radiation used (?.), the rays 
will fit together crest to crest, or in other words, the rays 
will be in phase and they will reinforce each other : i.e. 
in the direction AM there will be reflection. This is 
said to be reflection of the first order. Similarly, when 






DETERMINATION OF WAVE-LENGTH 21 

for another angle 6' the distance 2d sin 6' is equal to 
2/1, there is again reinforcement and the reflection is 
said to be of the second order, and so on. Generally 
we have 

2d sin = nX . . . . (1) 

where n = 1 for the first order, n = 2 f or second order, 
and so on. For the same conditions the rays scattered at 
A', 0', P', Q' reinforce in this direction. It may also 
be seen that the reflection is for one wave-length only. 
Of course it may happen, if a heterogeneous beam were 
used, that the angle is appropriate to the first order 
of one wave-length and the second order of a wave- 
length of about half the size : also, using a heterogeneous 
beam, it is apparent that each of the constituents is 
reflected when the appropriate angle of incidence, and 
the same angle of reflection, are used. For a given A, 
the intensity of the beam is very small in a direction 
not quite coincident with AM, due to interference of 
the rays caused by the extra path difference introduced. 
In other words, the lines are sharp. 

The X-Ray Spectrometer 

In the original Bragg spectrometer, a crystal of the 
kind considered in the preceding paragraph was mounted 
with its planes vertical on the prism table of a spectro- 
meter, as at C in Fig. 10. An incident beam of X-rays 
was limited by two slits, A and B, and the telescope of 
the instrument was replaced by an ionization chamber 
L The angle between the incident direction and the 
line from the crystal to the ionization chamber as limited 
by the slit D was always maintained twice the corre- 
sponding angle of glancing incidence at the crystal sur- 
face, in order to maintain the glancing angle of incidence 
and reflection the same, as in Fig. 9. 

The first experiments were made with rock salt, and 
the appearance of the curves showing the relation be- 
tween the ionization and the glancing angle is seen in 
**g- 11. As may be seen, ionization was found for all 



22 



X-RAYS 



DETERMINATION OF WAVE-LENGTH 23 



angles of incidence (all wave-lengths), but there is a 
noticeable feature in the comparatively sharp rises at 
certain definite angles of incidence, as, for example, at 
Aj, B lf Cj. The rises reoccur in similar proportions, 




Fro. 10 

but separated by about twice the former amount at 
A 2 , B 2 , C 2 , and there is further evidence of a thirl? 
appearance of these peaks. 

This showed that the radiation from the tube was! 
not evenly distributed among the wave-lengths, but 
that corresponding to the angle 6 having the approximate 




values 10°, 11° 30' and 14° there was an excess of 
radiation. The repeated form, A, B 2 C 2 , occurs at angles 
whose sines are double the sines of the angles corre- 
sponding to Ax, B 1} Cjl ; in other words this radiation was 
3 



24 



X-RAYS 



reflected in the first and second orders and the relation 
between the sines verified expression (1) given above. 
When the crystal was turned so that a Becond cleavage 
face was employed, the form of the ionization curves 
was as shown in II of Fig. 11. 

The conclusion that the peaks were due to the radia- 
tion was verified by using other crystals. The same 
form of ionization curves resulted, although the actual 
positions of the peaks were displaced to places which 
depended on the new lattice constant of the crystal 
used. Using X-ray tubes with different anticathodes 
and one crystal, the peaks were again of the same 
general form but in very different positions correspond- 
ing to other wave-lengths being emitted by the anti- 
cathode. This showed that these elements gave out an 
excess of certain favoured wave-lengths. The sugges- 
tion arose that these might be the characteristic rays 
which had already been found by absorption measure- 
ments — a conclusion which was at once verified by 
simple absorption tests. The ' peaks ' were, in fact, 
shown to be components of the K and L series, and 
may be called the emission lines of the material of the 
anticathode. Incidentally this showed that the ' homo- 
geneous ' rays were in reality a mixture of rays forming 
a narrow band of wave-lengths. Bragg investigated in 
this way the emission lines of Pt, Os, Ir, Pd and Rh. 

In order to determine the wave-length corresponding 
to any angle of reflection, it is necessary to know d, 
the lattice constant of the crystal. To understand the 
method underlying the determination of this constant, 
it is advisable to have at least an elementary knowledge 
of the terminology and the simpler features involved in 
crystallography. 

Crystal Structure s 

If, as indicated at the end of the last chapter, we 
consider a crystal to be made up by a constant repetition 

1 For a fuller account of this, see X-Ray Crystallography, by 
R. W. James, in this series. 



1. 



DETERMINATION OF WAVE-LENGTH 25 



of a fundamental unit, there is nothing in the crystal- 
lography of the case to decide the nature of the entities 
of which the unit is made ; they may be atoms, mole- 
cules, or groups of molecules. X-ray analysis leaves no 
doubt about this. We may take it that we are dealing 
with atoms. 

In Fig. 12 is shown a lattice unit whose sides are p, 
g and r. When this unit is repeated to make up the 
angles, it is clear that there are planes of atoms in the 




Fig. 12 

crystal parallel to OBFC separated by a distance p, 
parallel to OAGC separated by a distance q, and 
parallel to OADB separated by a distance r, and in 
addition there are obviously atom planes such as ABC, 
CBDG, etc., with other separations. 

It is found convenient to refer to these planes in 
terms of the intercepts on three axes. In the case of 
the cubic crystals the axes are at right angles, and 
p = q = r. The axes chosen are parallel to the cube 
edges : the ratio p:q:ris called the axial ratio. If we 



26 



X-RAYS 



take a plane XYZ passing through the atoms it seems 
clear that 

OX/p : OY/q : OZ/r = a:b:c, 
where a, b and c are whole numbers. For the simple 
cubic crystal, when XYZ is parallel to ABC it is evident 
that a : b : c = 1 : 1 : 1. 

The more usual ratio to take to define the plane is 
the reciprocal of the above, viz. : 

p/OX : q/OY : r/OZ = bc:ca:ab = h: k :l, 
where h, k and I are again whole numbers and are 
usually written in brackets, viz. : (h, k, I), and are 
called the indices of the plane in question. For example, 
a plane parallel to ABC is the (111) plane ; ADEG is 
the (100) plane, as the intercepts on the y and z axes 
are infinite ; BDGC is the (Oil) plane, and so on. 
When the crystal grows with faces such as (100), (010), 
etc., it becomes a cube, and is said to be of the form 
{100}. 

Determination of d and X 

The problem of the determination of d and X is one 
of finding two unknowns when only one relation, (1), 
is available. 

Fortunately the palladium anticathode used in the 
experiments emitted a strong characteristic line, and it 
was by a close study of the distribution of intensity of 
this line in the different orders of reflection from the 
various planes of NaCl and KC1 that the Braggs were 
able to make an estimate of the actual placing of the 
atoms, and hence the mass within a unit of the lattice. 
Direct calculation from a knowledge of the density gave 
a second estimate of this quantity, and so the value of 
d was obtained, as is seen in the following account. 

The determination of d was first undertaken by Bragg 
of the cubic crystals Rocksalt (NaCl) and Sylvine (KC1). 
Cleavage faces were made parallel to three planes, and 
the ionization curves obtained by reflection of the radia- 
tion from a tube with a paladium anticathode at these 









DETERMINATION OF WAVE-LENGTH 27 

faces are shown in Fig. 13. The first order reflection 
from the (100) plane of KC1 is at 20 = 10° 43' and for 
NaCl at 20 = 11°48'. Now equation (1) gives 2d sin 0=A 
for the first order, and therefore if these crystals are 
simple cubic in character with a lattice unit of side 
dj and [d 2 respectively we have 2d, sin 5° 21' = X = 



Sy/<""e 
KG 
















(lOO) 


Tj 




- .- A 








(HO) 




.11 










i in ) 






^_/L 














| 












(100) 


I 




n f\ 








(no) 




r 


JL 










tin) 






_*.!_ 









0' 



5' 10' 

Pd Roy 



13' 



to' 



SO' 



36' 



Fig. 13 

Angulnr setting of Chamber. 

2d 2 sin 5°-9, or <£, = 5-48 X ; d s = 4-85 X, where X is the 
wave-length of the prominent palladium line used. 

Again, from general considerations it seems reason- 
able to suppose the molecular volumes of the crystals 
are in the ratio of the cube of the corresponding d, and 
jf M and p are the molecular weight and density we 
have 



28 



X-RAYS 



mol. vol. KC1 : mol. vol. NaCl = M,/p, : M 2 /p 2 = d x * : d, 8 
or d 3 1 p l /M l = (Z 3 2 p 2 /M 2 = a constant. 

For this type of crystal, therefore, one anticipates the 
constancy of dtyp/M. The results actually obtained in 
terms of X, for three crystals are : 
d-VJ/E = KC1, 1-63 % ; NaCl, 1-62 X ; KBr, 1-63 A, 
which points to a similar structure in these cases. 

In order to decide the exact form of the lattice unit 
we will next consider the spacing of atom layers for 
crystals made up of the units shown in Fig. 14, where 
(a) is a cube of side a with diffracting centres at the 
corners, (b) is a face centred cube of side 2a and (c) is 
a cube centred unit of side 2a. The reason for selecting 
a and 2a in the different cases will appear later. 

In these three cases the actual spacing of the planes 
is found by simple calculation to be : 






(100) 
Simple cube lattice (a) a ... 

Face centred ,, (6) a ... 

Cube centred „ (c) a ... 



(H0)_ 

o/V2. 

a/\/2. 
2a/ VI. 



(Ul)l 

. a/V3 

.2a/ V 3 
. a/VT I 



The next step, therefore, is fairly clear ; a ratio of 
the spacing of the various planes must be determined. 
For KC1, as shown in Fig. 13, the angle for the first 
order reflection at these faces has the values 5 0> 22, 7°-30, 
9°-05, i.e. 
d(ioo) : d( llQ ) : d( in ) = 1/sin 5-22 : l/sin7-30 : l/sin9-05 

= 1 : 1/vT: 1/V3T 

which agrees with the simple cubic system. On the 
other hand, if we consider NaCl we find identical results 
only if we neglect the weak reflection at the (111) face 
at about 5°. When we take this weak reflection into 
account we obtain 

d(ioo) : d( uo ) : d( ni ) = 1 : 1/vT: 2/\% 
which corresponds to the face centred system. 
The solution of the problem as given by Bragg is 








^ cS "* 




30 



X-RAYS 



illustrated in Fig. 15, which represents either NaCl or 
KC1. The black dots show the positions of the CI atoms, 
and the open circles either Na or K atoms. It will be 
seen that they form an interlocked face centred cube. 
In the case of KC1 we are dealing with atoms of atomic 
number 19 (K) and 17 (CI), which in combination have 
each 18 electrons, so that they each scatter to the same 




^ — a- — 



NeO NoCI 



-.?. 



NoCJ fihOAbO 



Na CI No a Na 



tr 








i 
i 

i 


i 



too planes 



(no) 

Fig. 15 



a "/y/3 («') 



extent and from this point of view are indistinguishable : 
the system of scattering centres therefore behaves as 
though made up of a simple cubic unit of side a. 

In the case of NaCl we have atoms of atomic numbers 
11 (Na) and 17 (CI), which in combination have 10 and 
18 electrons respectively. As shown in the lower part 
of Fig. 15 the (100) and (110) planes are similar in com- 
position, being made up of an equal number of Na and 
CI atoms, but the alternate (111) planes are made up 



DETERMINATION OF WAVE-LENGTH 31 

entirely of Na and of CI atoms. If we reflect at this 
face, the CI atoms will act as a face centred cube of 
side 2a, and if we could neglect the reflection from the 
Na atoms, the resulting reflection would be appropriate 
to a face centred cube. However, midway between the 
CI planes are the Na planes which also give rise to a 
reflected beam. When the conditions are right for the 
first order reflection at the CI planes the corresponding 
path difference for the reflection at adjacent planes, 
i.e. at alternate Na and CI planes, is exactly that for 
destructive interference. H the intensity of reflection 
from the atom planes were the same, there would be 
no resulting intensity. But the CI planes (18 electrons 
per atom) reflect more copiously than the Na, and the 
result is a reflection appropriate to the difference of the 
intensities, and a weak line is produced, as we have seen. 
The scheme of Fig. 15 therefore satisfies the observed 
results. 

In order to find d, it must be realized that when 
the lattice unit is repeated throughout the crystal, the 
corner atoms are common to eight cubes, viz. A, B, C, 
D, etc., are each common corner atoms to eight cubes, 
and since there are eight such corner atoms in each 
cube, only one is to be regarded as proper to each large 
cube. In the same way the face centering atoms are 
each common to two cubes, and there are six such 
atoms, K, L, M, N, O, P, with a result that there is 
a contribution of three atoms from this source to the 
total content of the large cube unit, which has there- 
fore a total of four atoms of this kind. A similar reason- 
ing shows that the other atom makes an equal contri- 
bution to the cube, with which we must consequently 
associate four molecules. The side of the cube is 2d, 
and therefore the mass within it is (M) 3 p, where p is 
the density of the crystal. But if m is the mass of the 
hydrogen atom, each molecule has a mass Mm, or from 
X-ray data the mass of the cube is 4Mw, 

i.e. d 3 = £Mw/p. 
For rocksalt Bragg made the following substitution : 



32 



X-RAYS 



M = 29-25, m = 1-64 x 10" 24 , p = 2-17, which 
d = 2-8 x 10" 8 cm. and X rd = -576 10" 8 cm. 1 

Having found the spacing for one crystal in this way, 
the wave-length of reflected X-rays can be calculated and 
the lattice constant of other crystals may be evaluated. 
In this manner the foundation of ray examination of 
crystal structure and of X-ray spectroscopy was laid. 

Moseley's Law 

In the first spectrum as illustrated in Fig. 11, it was 
seen that the L series of palladium was made up of 
three distinct components which were separated by 
about 1°, i.e. the L radiation was not truly homogeneous, 
but was a group of radiations. A systematic search into 
the characteristic radiations of some thirty-eight ele- 
ments was undertaken by Moseley in 1913. He found 
that both the K and L series could be separated in this 
way into a system of lines in all cases. In general the 
K series consisted each of two lines ; the longer wave- 
length, and more intense line was called the K„, the 
other the Kp line. In the same way the L radiation 
from each element was found to be made up of three 
lines which were called L„, L^, L y , in order of decreasing 
wave-length and intensity. The spectra of the elements 
were of the same general character, but varied in wave- 
length in a very regular manner. Moseley found that 
the regularity was most pronounced if he plotted the 
square root of the frequency of corresponding com- 
ponents against the atomic number of the source, rather 
than the atomic weight. (In view of the Rutherford 
atom, the atomic number may be regarded as the num- 
ber of free positive charges in the nucleus.) Moseley's 
work emphasized the importance of atomic number (N) 
and may be said to have established its real significance 
and use in physics. Fig. 16 shows a typical diagram 
giving Moseley's results, in which it will be seen that 
the linear relation between v' (frequency) and N is very 

1 Alternatively if L is the number of molecules per gramme 
mol. (606 x 10"), 8 d*p = 4M/L whence d = (M/2pL)». 



r 

gives del 



DETERMINATION OF WAVE-LENGTH 33 

definite ; in fact, he was able to predict in a few cases 
that new elements were to be expected in places where 
a departure from this regularity was noticed in a pre- 
liminary plot. One of these elements has since been 



WovtkngFh 



<lAg 



SBSr 




Square 



Frequency 
Fig. 16 



Relation between the Atomic Number of the source and the square root ol the 
frequency of the radiation. 

isolated by Bohr and called Hafnium (N = 72). Con- 
firmation was also to be had from his results that the 
atomic number should be in the reverse order from the 
atomic weight in certain pairs of elements whose chemical 
properties also demanded this reversal (e.g. cobalt and 
nickel, and tellurium and iodine). Incidentally these 
results provided a comparatively rapid means for the 
analysis of substances which otherwise require a lengthy 
and involved chemical process. 



34 



X-RAYS 



From the point of view of spectroscopy the important 
feature of Moseley's work was the fact that the frequency, 
v, of the radiation in the corresponding components of 
both the K and L groups was simply related to the 
atomic number in a way which is expressed in the 
following equation : 

v = A(N-6) 2 .... (2) 
where A and b are constants for each set of lines. This 
result, which is known as Moseley's Law, was very soon 
shown to be consistent with Bohr's theory. The funda- 
mental importance of the work warrants a passing con- 
sideration of this theoretical aspect. 

Bohr's Theory 

As is well known, Bohr was able to deduce expressions 
for the line series of hydrogen, by making use of the 
Rutherford atom model. (See, for example, Spectra, 
by Johnson, in this series.) 

The so-called Rutherford-Bohr atom consists of a 
nucleus of charge (-f Ne) surrounded by N electrons 
which move in orbits of different radii about the nucleus, 
called stationary orbits. The possible orbits are deter- 
mined by postulates which are set out later in this 
section. It is assumed further that the electron may 
rotate in a stationary orbit without radiating. Of course 
according to the classical mechanics whenever an elec- 
tron is accelerated it radiates : when moving in a circular 
path it is always subjected to an acceleration towards 
the centre, and therefore should be continually radiating 
and moving in an orbit of continually reduced radius. 
This leads to results which are very different from the 
observed events and led to the Bohr theory which is 
outlined in a simplified form in this section. 

The innermost orbit in the atom is called the K orbit 
and the corresponding energy the K ' level ', the next 
is the L orbit, and so on. When the atom absorbs 
energy from radiation, or by the incidence of a quickly- 
moving electron (as in an X-ray tube), the energy may 






DETERMINATION OF WAVE-LENGTH 35 

be used in the ejection of an electron from one of the 
stationary orbits against the electrostatic attraction of 
the nucleus. The vacant space is filled up by the fall- 
ing in of an electron from one of the outer positions, 
and the work done by the electron as it falls in appears 
as radiation. If the space filled up is in an outer 
orbit the radiation may be in the visible region ; if a 
K electron is replaced, the radiation is the K radiation 
of the atom. The final position of the electron deter- 
mines the type of radiation, and the starting position 
of the electron, e.g. the L, M, N, etc., ring, or orbit, is 
responsible for the structure of the radiation, e.g. K tt) 
Kfl, Ky , etc. We conclude, therefore, from this theory 
that the characteristic radiations are caused by the 
jump of an electron from one stable orbit about the 
nucleus to another nearer the nucleus. If an electron 
moves in an orbit of radius a, it is assumed : 

(1) That the electrostatic attraction of the nucleus 
(of charge E = Ne) balances the centrifugal force mv z /a ; 
or 

Ne*/a* = mv*/a .... (2) 

(2) That the possible orbits are those in which the 
moment of momentum is an integral multiple of h/2n 
(where h is Planck's universal constant), or 

mva = nh/2jt (3) 

where n is a whole number. 

(3) That when an electron jumps from an orbit of 
radius a u having an energy W„ to one of radius o 2 with 
an energy W 2 , the frequency, v, of the radiation emitted 
is given by 

/ iV = Wx-W (4) 

The term W is the total energy of the electron, i.e. 
hnv* -f the potential energy. The potential energy of 
the electron as measured by the work done on it in 
bringing it from a place of zero energy, i.e. an infinite 

distance away, is Ne.e/r 2 .dr = — Ne 2 /a. 



30 



X-RAYS 



Substituting the values of a and v from (2) and (3) 
we have 



W = - 2jth i N 2 m/n 2 h 2 



(5) 



In the case of any atom other than hydrogen the 
expression for the attraction as given in (2) must be 
modified because the remaining N — 1 electrons exert 
a repulsion on the one considered. For example if 
there are N x electrons within the orbit, they behave as 
though concentrated at the centre and produce a repul- 
sion Nje.e/a 2 ; if there are N 2 electrons moving about 
orbits with the same radius, a, they exert a repulsion 
on the electron considered which is a simple function, /, 
of N 8 e 2 /a 2 ; the remaining electrons have no effect on 
those with in their path . The net electrostatic attraction 
is therefore 

Ne 2 /a 2 - N,eVo 8 - /(N 2 )e 2 /a 2 
= (N-6)e 2 /a 2 

where b = N!+/(N a ) and is small for the inner electron 
orbits. The amended value of W is therefore 



W=- 



27i 2 e*m(N - 6) 2 
n 2 h* 



and for a transition which results in a radiation 
27i*e*m { /N - 6,\ 2 /N - 6a 2 ' 



V as 



£ 3 



1(^)'-(^ 



Since we are considering the K and L rings, which are 
the two inner rings, the value of b is small in each case 
and we may write 



■*»*-*&-£' 



h- 



?i 



-**-"*{&-&} 



where R is Rydberg's constant. 

When we consider the production of corresponding 






DETERMINATION OF WAVE-LENGTH 37 

lines in different elements the transitions are the same, 

and therefore we may put A = R j — 5 21 anc * 0Dtam 

v = A(N - 6) 2 
which is identical with Moseley's Law. 

Development of Technique 

The original Bragg spectrometer, in which an ioniz- 
ation chamber was used, has been described. In the 
Moseley experiments a photographic method was used. 
A slightly diverging beam of X-rays was allowed to 
fall on a crystal, and the corresponding reflected rays 
formed images on the photographic plate appropriate 
to a small range of glancing angles which was deter- 
mined by the divergence of the beam. The photographic 
plate, therefore, was useful for a range of wave-lengths, 
and the whole spectrum was mapped out in small stages 
by successive small rotation of the crystal. 

The spectrometer used by the Braggs in later experi- 
ments was automatically adjusted by a clock-work 
mechanism which moved the ionization chamber through 
twice the angle of the crystal. They also introduced 
a focussing device which consisted in placing the slit of 
the ionization chamber at the same distance from the 
crystal as the slit nearest to the source of X-rays. This 
enabled wider slits to be used and reduced exposure 
time. 

M de Broglie introduced a photographic method where- 
by the whole spectrum was recorded on a photographic 
plate by clock-work rotation, to and fro, of the crystal. 
This model has been used as a basis for a development 
of a spectrometer which, in the hands of Siegbahn and 
his school, has proved an instrument of the highest 
degree of precision. This point may be probably realized 
if one considers the order of measurement obtainable. 
In the visible spectrum the Angstrom unit (10~ 8 cm.) 
is employed, but for X-rays, Siegbahn has introduced 
a new unit — the X unit — which is 10" 11 cm. With the 



38 



X-RAYS 



last apparatus referred to, measurements to -01 of an 
X.U. are obtainable, i.e. Siegbahn's measurements are 
probably correct to 10" 13 cm. There is reason to believe 
that in these measurements consistent values of wave- 
length are now available which for their absolute 
accuracy depend only on the value of d, the lattice 
constant of the crystal used. This, as we have seen, 
depends upon a knowledge of a mass of the hydrogen 
atom or, what comes to the same thing, the value of 
Avograda's constant, the number of atoms per gram- 
molecule. Unfortunately the evaluation of this con- 
stant is not yet carried out with the same extreme 
accuracy, and an accepted value for d is agreed upon 




Fio. 17 

for certain crystals as the most probable in the light 
of recent experiments. 

The developments just considered whereby a high 
degree of precision is obtainable gives means of accurate 
wave-length determination. By photometric analysis of 
the plate relative intensities may also be found ; also 
for the measurement of intensity of lines, a development 
of the ionization methods is very often utilized. An 
example of this kind of thing is the self-recording ioniza- 
tion spectrometer which Compton used for the measure- 
ment of the intensity of spectral lines. The results of 
an investigation with this type of apparatus are of the 
form shown in Fig. 17. The approximately horizontal 



DETERMINATION OF WAVE-LENGTH 39 

line in that figure shows the variation in the intensity 
of the total incident beam, as measured in small ioniza- 
tion chamber fixed in that beam, the other line gives 
a measure of the intensity at the different angles shown. 
It is measured by the ' steady deflection ' set up in a 
quadrant electrometer which is connected to the ends 
of a very high resistance, which forms a leak to earth 
from the ionization chamber which moves about the 
crystal. 

Limitation to the Use of Crystals 
Later work has investigated a large range of wave- 
lengths but, from the nature of things, the crystal 
spectrometer method must break down for extreme 
wave-lengths. For example, it is obvious that for an 
X-ray whose wave-length is equal to 2d, the first order 
reflection will be at 90° and for a radiation of longer 
wave-length no first order image will appear. The 
examination of wave-lengths of this order is beset with 
other difficulties very largely due to the fact that these 
rays are very highly absorbed in a few centimetres of 
air. To work in this region the whole spectrometer is 
placed in an evacuated space. Such a vacuum spectro- 
meter is used in the investigation of the M and N rays, 
using certain inorganic crystals which have a large lat- 
tice constant. In the other extreme with short X-rays 
and y-rays, the lattice constant is too coarse in all 
crystals, and special methods have to be devised as, for 
example, those of Rutherford and Andrade and those 
methods which depend on photoelectric emission. 



CHAPTER in 
THE SCATTERING OF X-RAYS 

Thomson's Theory of Scattering 

THE classical method of regarding the genesis of 
X-rays, as briefly mentioned on page 13, was one 
in which the radiation was imagined to be set up as a 
result of the retardation of the electron in the parent 
cathode stream. On these lines Sir J. J. Thomson 
made a theoretical estimate of the scattering to be 
expected when X-rays fall on matter. His method was 
to calculate for electrons in the scattering substance 
the acceleration produced by the electric vector in the 
incident primary X-ray beam. Then, assuming that 
the accelerated electron radiates according to the 
classical mechanics, he deduced the intensity of the 
scattered radiation in any direction inclined to that of 
incidence. His calculation gave the intensity of the 
scattered beam 1$, in a direction inclined at 6 to the 
primary beam and in a plane containing that beam, 
to be 

1.-5^1 + 008*6) 

If we put = 7i/2> we have intensity of the scattered 

radiation at a distance r from the scatterer and in a 

Iwe* 
direction inclined at 90° to the incident is Itta = . . , - or 

2raVr z 

Io = W 2 (1 + cos 2 0) . . . (1) 
where n is the number of scattering electrons, e and m 
are the charge and the mass of the electron, c is the 

40 



THE SCATTERING OF X-RAYS 



41 



velocity of the radiation, and I is the intensity of the 
incident beam. 

If we integrate the expression over a complete sphere 
we obviously obtain I„ the total energy scattered in all 
directions, viz. : 

I, = I l e .2nr sin d.r.dd 



- 



I.= 



Slnne* 
3m 2 c 4 



which is, of course, independent of r and gives as the 
value of the fraction of energy scattered, 8, 

*' 8 we * " ... (2) 



3 m 2 c* 



.v. 



It is to be noted that the fraction of energy scattered, 
according to this deduction, is independent of the 
wave-length of the radiation and merely depends on 
the number, n, of scattering centres (electrons). The 
underlying assumption is that the n electrons are free 
to scatter independently of each other, and the total 
is n times the scattering per electron. By the nature 
of the theoretical treatment it is also assumed that the 
scattered radiation has the same wave-length as the 
incident beam. These results were obviously capable 
of experimental verification, and in 1911 Barkla and 
Ayres, also E. A. Owen and Crowther, carried out 
experiments with this in view. 

Distribution of the Scattered Radiation 

The method of attack in these early experiments 
was to irradiate elements of atomic weight less than 
that of sulphur. The characteristic radiation of these 
elements is very soft, and is absorbed in a few centi- 
metres of air at normal pressure. Also the photoelectric 
emission is similarly absorbed, so that it is clear that 
any radiation received in an ionization chamber some 
10 cm. away from the light element is almost entirely 
scattered from that element. 



42 



X-RAYS 



A scattering coefficient s was introduced, such that 
in cases of loss to the beam by scattering only, in a 
path x cm. long, the relation I = I e _,x defined the 
*. Corresponding to this the mass scattering coefficient 

is 8/p. 

It was found that the equation (1) above represented 
the distribution of intensity, except for the forward 
directions, where, especially for small angles, the 






THE SCATTERING OF X-RAYS 



43 




iM *&a 



"• Scoirenng Aivjle — 

Fig. 18 

difference between (1) and experimental values was 
appreciable. There was a general agreement in the 
rise and fall of the radiation with 0, but not an exact 
numerical one. The radiation used was a filtered 
radiation from a tube, but was by no means homo- 
geneous. 

More recently this point has been re-investigated by 
Hewlett (1922), using an incident beam of monochro- 






matic radiation of wave-length -7lA.U. Hewlett en- 
deavoured to reproduce the theoretical conditions of 
Thomson by using a liquid scatterer of low atomic 
weight (mesityline — C 6 H 3 (CH 3 ) 3 ) where it was pre- 
sumed that the distribution of electrons is such that 
they scatter independently, as was assumed in the 
development of the formula. The results of this 
investigation are shown in Fig. 18, where it will be noted 
that there is the same departure from the theoretical 
curve as was obtained in the early experiments, but it 
is also clear that there is a very marked decrease in 
the scattering for angles less than 10°. This decrease 
at small angles is comparable with results obtained by 
Keesom and Smedt, who passed a narrow beam of 
X-rays through liquid oxygen, neon, nitrogen, carbon 
dioxide and water, and received the beam on a 
photographic plate. It was found that the photo- 
graphic image showed a central maximum surrounded 
by a ring of minimum intensity, which in turn was 
followed by a halo-like ring of maximum intensity. 
The first minimum corresponds closely with the 
small intensity shown in Hewlett's experiment. 
Keesom and Smedt explained their results in a very 
satisfactory way in terms of a theoretical treatment 
given by Debye. This was a calculation of the resulting 
scattering produced by pairs of scattering centres 
separated by a fixed distance, s. Such centres were 
shown to produce diffraction rings. It was then assumed 
that the pairs of scattering centres in the case of the 
liquids used were the bi-atomic molecules, and the 
distance s was found to be in agreement with the distance 
between the molecules when closely packed. In other 
words, the good agreement which they obtained made 
it clear that in a liquid it is not safe to assume that 
there is a perfectly random arrangement of the mole- 
cules. It was pointed out by Compton that the mini- 
mum in the experiments of Hewlett was, in the same 
^ay, the result of interference of the radiation scattered 
by the neighbouring molecules. It was shown, in 



44 



X-RAYS 



fact, that the position in the case of the small angle 
scattering was similar to the reflection of X-rays at a 
crystal for angles less than that corresponding to the 
first order. The reinforcement at the larger angles 
was sufficient to explain the excess scattering observed. 
With this explanation for the apparent deviations, it 
became clear that the classical theory was capable of 
accounting for the observed intensity distribution, in 
the case of X-rays of long and medium wave-length, 
coming from low atomic weight elements. In the case 
of very short wave-length radiations (y rays from Ra C 
X = 02 A.U.), Compton showed that the distribution 
was very different from the classical one except for 
small angles. He was unable to obtain a theoretical 
counterpart for his results except by having recourse 
to a quantum theory (see page 47). 

The Scattering Coefficient 

It was seen that equation (2) gives the energy scat- 
tered by n electrons. If we let n be the number of electrons 
per cc., then s has the same meaning as given on page 42, 
because I refers to unit cross section, and therefore 8 
refers under these circumstances to the scattering per 
unit length of path ; also the definition of 8 on page 42 
is the equivalent of writing the fractional loss in energy 
in a path of length dx in the form 

dl/I =-s dx 
and when the electron distribution is uniform the 
fraction lost per cm. is clearly equal to s. 

Experimental work in this field has yielded results 
as interesting as in the last. For example, in the work 
of Barkla and Sadler (1911) already quoted, a/p was 
found to be 0-2. When this value is substituted in 
equation (2), the number of electrons per gramme, 
and hence the number per atom, may be calculated. 
This gave a value of N (the number of electrons per 
atom) approximately equal to half the atomic weight 
of the scatterer, and incidentally provided the first 



THE SCATTERING OF X-RAYS 



45 



reliable estimate of the number of electrons to be 
associated with an atom. Since at the present time 
this number is well known from other considerations, 
we may look on the result as showing that the scattering 
from low atomic weight elements is produced by all 
the electrons outside the nucleus, acting independently 
when fairly long wave-length radiation is used. In the 
later experiments of Hewlett where monochromatic 
radiation was used the value of the scattering coefficient 
obtained confirmed Barkla's earlier result, viz. 0-2, and 
on substituting the modern atomic and electronic 
constants a more exact value of N was obtained. 
[N = 6 for carbon from the experiment quoted in the 
last section]. 

H we accept the classical theory, therefore, the value 
of the mass scattering coefficient is 0-2 for all wave- 
lengths. 

Experiment has shown, however, that for long wave- 
length radiation (soft X-rays) and high atomic weight 
scatterers there is an excess of scattering above that 
predicted by theory. In this case there is no real 
contradiction of the theory, it is merely due to the 
experiment not conforming to the conditions of the 
calculation. Electrons do not scatter independently 
when the wave-length of the incident radiation is great 
compared with the distance between them. It has 
been shown that under these circumstances the elec- 
trons in one atom scatter as a group, and that in the 
extreme case the scattering to be expected, assuming 
classical laws, is proportional to N* and not to N. 

A real departure from the results anticipated by the 
classical theory is found in the case of the scattering 
of short wave-length radiation. In all experiments 
on scattering of X-rays of wave-length less than 0-3 
A.U., s/p falls below the predicted value of 0-2, and 
in the case of y rays the mass scattering coefficient is 
found to be as low as 0-05. These results cannot be 
reconciled with the older theory, but as in the case of 
the distribution mentioned in the last section, quantum 



46 



X-RAYS 



theories have been proposed which appear to fit the 
results quite well. Some account of these will be 
given later. 

Wave-length of the Scatteeed Rays 
In the early experiments where care was taken to 
use long wave-length radiations and low atomic weight 
scatterers, simple absorption measurements on the 
scattered rays showed that, within experimental error, 
the absorption coefficient was the same as for the 
primary beam, i.e. that the wave-length was the same, 
as the theory had predicted. However, from about 
1904 onwards there had been evidence that under other 
conditions the scattered rays were softer than the 
incident beam. In 1917 Barkla sought to explain this 
effect m terms of a new homogeneous radiation from 
the scatterer, which he called the ' J ' radiation, because 
it appeared to be more penetrating than the K radiation 
of that substance. This J radiation would account 
for the observed softening, as a homogeneous radiation 
is always of longer wave-length than the incident 
radiation which excites it. Again, in 1922 Crowther 
suggested that the softening which he observed in his 
experiments could be explained either by the presence 
of a J radiation, or by a possible increase in wave-length 
of the scattered beam. He decided that the former 
explanation was the more probable. 

From theoretical considerations it was of the utmost 
importance to decide whether this J radiation was 
present. As seen on page 34, there is no place of origin 
m the accepted atom models for a more penetrating 
radiation than the K, as these rays are supposed to 
be due to the falling of an electron to the innermost 
ring. The matter was finally settled by the experiments 
of L)uane and Shimizu, who made a careful investigation 
of the emission lines of aluminium, the element in which 
the supposed J radiation had been observed. An X-ray 
tube with an aluminium anticathode was used, and the 
spectrum of the radiation was recorded over a wide range 






THE SCATTERING OF X-RAYS 47 

of wave-lengths. No trace of the J radiation was found. 
This suggested that the results of Barkla's experiments 
had been misinterpreted. At the same time Barkla also 
came to this conclusion as a result of other experi- 
ments, and so the J radiation as such was abandoned. 1 

In 1923 A. H. Compton investigated the other possi- 
bility mentioned above, namely that of a wave-length 
change on scattering. This resulted in a most remarkable 
theoretical and experimental treatment of the subject. 

Compton's Theory 

Compton's treatment of the problem on the theoretical 
side is based on the quantum theory. Before giving 
an outline of his theory, it is perhaps advisable to refer 
to what will appear in a later chapter, and to remind the 
reader of one or two consequences of the application of 
the principles of relativity to the quickly moving electron. 

On pages 35, 64, and 74 are to be found specific instances 
from X-ray data, and in Johnson's Spectra, Chapter I, 
more general considerations are given which led to the 
formulation of the quantum theory. 

It is a fundamental postulate of the theory that when 
there is an interchange of energy between matter and 
radiation, it always takes place in integral multiples 
of a fundamental unit or quantum, which, for a radia- 
tion of frequency v is hv, where h is a universal constant 
(equal to 6-55 X 10" 27 erg-sec), called Planck's constant, 
i.e. there is no gradual interchange of energy. 

The principle of relativity leads one to associate with 

1 Continuously from that time Prof. Barkla and his school 
have made a careful study of an intermittent softening which 
they find in scattering experiments. It has been claimed that 
the scattered rays are of the same penetration as the primary, 
and that the apparent change is produced in the absorbing 
screens. This effect is called the J phenomenon. The scattered 
beam is found by them to be modified when tested in one sub- 
stance, and unmodified when tested in another. Other experi- 
menters have not yet been able to reproduce the effect. 

A summary of those experimental investigations is to be found 
in Science Progress, Oct. 1928. 



48 



X-RAYS 



the energy of the quantum, hv, a momentum, hv'c 
where c is the velocity of the radiation. 

Again, the application of the principle of relativity 
shows that if m is the mass of an electron at rest, it has 
a mass equal to m/Vl - /S 2 (where /? is v/c) when 
moving with a velocity v. The same principle shows 
that the kinetic energy of the electron is 

mc*(l/Vl - p* - 1). 
It may be seen that when v is small, this expression 
reduces to W, and the expression for the mass becomes 
equal to m. 

Compton visualized the quantum hv as a concrete 
centre of energy, in a way reminiscent of the cor- 
puscular theory of light, and considered what would 
happen when such a quantum came into collision with 
a stationary free electron. In the case of the emission 
of photoelectrons we assume that the whole of the 
energy of the quantum is given to the electron (as is 
seen on page 74), but in the case of the free electron 
of the present problem it was assumed that the ordinary 
laws of impact of elastic spheres would hold. 

In Fig. 19 a quantum of energy in a beam of frequency 
v incident m the direction 10, meets a stationary electron 
at O. The result is that the electron moves off in a 
direction OP with a velocity v and the residual quantum 
reduced to some smaller value hv', moves in the direc- 
tion OS. 

Referring to the figure we see that the momentum 
ot the electron is in a direction inclined at to the 

incident direction, and is mv/Vl - p* ; — , the new 

momentum of the scattered quantum, is in a direction 
inclined at <f> to the incident direction. 

We may express the conservation of energy in the 
form of an equation as 

hv = hv' -f mc 2 ( — A 



THE SCATTERING OF X-RAYS 49 

Then, expressing the momentum in the line of incidence 

and at right angles to this direction two more equations 

are obtained : 

hv hv' , . mv 

— = — . cos <f> + 



Vi-p* 

mv 



n hv ' • j: i 

O = — . sin -f- . _ 



-. cos 

! 

. sin 



Solving these three equations we obtain Compton's 

s 

Scattered Quontum 




/ Incident Quantum 
Momentum by 



Recoil Ckctror 
Momenlitm 



Fio. 19 
The collision of an incident quantum hv' and a free electron, resulting in the 
recoil of the electron and the scattering of the residual quantum hv . 

result, for the wave-length of the scattered radiation 
A', corresponding to the frequency v' 

V = X + £- (1 - cos <f>) 
mc 

i.e. the change in wave-length to be expected on this 

theory is 

A'-A = -d-cos^) ... (3) 
mc 

It will be noted that the predicted change of wave- 
length has the following properties : ,„,.., 

(1) It is independent of the wave-length of the incident 
beam ; 



50 



X-RAYS 



Ma/ybc/enum 
ka Line 
Primary 



(2) it is independent of the nature of the scattering 
substance ; ^* 

(3) it varies with the angle of scattering. 

These results offer a direct 

challenge to experiment, 
and the work of Compton 
and his collaborators made 
a worthy counterpart to the 
theoretical investigation just 
summarized. 

Molybdenum Ka rays 
were directed on to a carbon 
scatterer and the rays scat- 
tered in a given direction 
with respect to that of inci- 
dence were limited by lead 
slits which acted as the slits 
of a Bragg spectrometer, 
which in turn served to 
measure the wave-length of 
the scattered rays. By mov- 
ing the X-ray tube round 
to direct the beam through 
the slits it was possible to 
obtain the plot of the inci- 
dent beam ; also by moving 
the tube the angle of scat- 
tering could be set at any 
value. Fig. 20 shows the re- 
sult of these investigations. 
The upper figure gives the 
plot of the incident beam 
about the position of the 
Ka line and the other figures 
7-30- show the effect on this line 
of scattering at different 
... ... r , angles. It is to be seen 

that the change of wave-length is unquestionable : the 
modified line is present, together with an unchanged 




THE SCATTERING OF X-RAYS 



51 



line in each case. It will be noted that the change in 
the position of the modified line increases with increase 
of the angle of scattering as anticipated. 

Fig. 21 shows the result of a second set of observa- 
tions (Woo) designed to test the second point mentioned 
above. It is seen that when a given angle of scattering 
is taken (120°) and different scatterers are used, that 
the change in wave-length is the same, but the relative 




Angular Setting of Spectrometer 
Fio. 21 

amount of modified scattered ray changes ; the higher 
the atomic number of the scatterer the higher the frac- 
tion of the unmodified ray ; with lithium the whole 
of the radiation is modified, with silver there is no 
modified ray. 

With regard to the first property of the change in 
wave-length mentioned above, namely that it is inde- 
pendent of the wave-length of the incident beam, it 
would appear that this change should occur in the visible 
region. Experiments which were made by Ross, on 
the scattering of mercury light by paraffin, made it 
clear that no such change take place. On the other 



62 



X-RAYS 



hand experiments with y rays show that all the scat- 
tered rays in this case are changed in wave-length 

Again using a fixed wave-length on different scat- 
tered the amount of modified radiation in Woo's 
experiments was seen to decrease as the atomic weight 
of the scatterer increased. Now, it was seen that the 
result of equation (3) was obtained by assuming that 
the quantum encountered a free electron, but it is clear 
that, if the electron is bound, and unable to move it 
will not take any energy from the incident quantum 
which proceeds without loss, i.e. the radiation is un' 
changed. This occurs either when the incident quantum 
is small as in the case of visible light, or when the binding 
of the electron is great for an incident quantum of higher 
value, but does not occur for the relatively large quan- 
tum associated with the y ray. We should therefore 
expect from this theory the type of result which is 
found in the experiments. 

The one final point about the theory which is to be 
mentioned, is concerned with the electron which en- 
counters the incident quantum, shown in Fig 19 as 
recoiling at an angle 0. This so-called recoil electron 
had already been observed in the cloud condensation 
experiments of Wilson and Bothe, and the theory at 
once gave a satisfying explanation of the short tracks 
as is shown on page 85. 

To account for the intensity of the scattered beam 
which we have seen is not explicable in terms of classical 
mechanics under all conditions of wave-length and 
atomic weight, various theories have been advanced, 
notably by Debye, Jauncey, Woo, Compton and Breit. 
lhe quantum theories of Compton and Breit appear 
equally well to fit the observed results. But a recent 
solution m terms of the new quantum theory by Dirac 
is identical with Breit's result, and at the same time 
Ihrac finds that equation (3) may be developed directly 
by this new method of attack. This equation has 
also been deduced without the assumption of Compton 
by the methods of wave mechanics by Schroedinger 






" 



CHAPTER IV 
EMISSION AND ABSORPTION 

Emission Lines 

WE have already seen that the early Bragg 
ionization curves for the reflection at the face 
of a crystal gave the emission spectrum corresponding 
to Barkla's K and L ' homogeneous ' radiations, showing 
these to be a group of lines in each case. These and 
other early spectroscopic investigations showed that the 
K series from all the elements were apparently a mix- 
ture of two lines and that the L series were a group 
of three lines. Later investigations have shown that 
the K lines are themselves doublets and that the light 
elements present a more complex spectrum, the addi- 
tional lines being sometimes called the ' spark ' lines 
by analogy with a similar case in the visible region. 
With modern high resolution, on the other hand, the 
L series presents at all times a spectrum with many 
lines, the complexity of which increases as the atomic 
number of the element increases. For both series it 
has been found that the width of the line increases 
beyond that set by the slit width, for radiation coming 
from the lighter elements. This apparently corre- 
sponds to the presence of a fine structure. 

The K Series 

In the K series the common form for all elements 
is that of two doublets. The first corresponds to 
Moseley's Ka line and is a very close doublet whose 
components a, and a a are separated by only a few 

53 



54 



X-RAYS 



X.U. : ' the second line of Moseley, the K0, splits up 
into /?! and /3 2 . The notation used denotes by a the long 
wave-length doublet and by (} the short wave-length 
doublet. The line of maximum intensity has a suffix 1, 
the next in order of intensity has a suffix 2, etc. All 
the points referred to are shown in Fig. 22, which 
is a drawing of photographs given by Siegbahn. It 
shows the gradual decrease in wave-length with rise in 



Directly transmllttd beam 



fitfl W 
























As 






Br 



fib 



V 






Fig. 22 

atomic number of the source and also indicates the 
relative intensities of the lines. When the relative 
intensities of the lines are measured, either by ionization 
methods or by photometric analysis of a photographic 
plate, it is found for a wide range of elements that the 
a x line is about twice as intense as the a 2 . The a group 
as a whole is several times as intense as the associated 
/9 group. 

In some cases in the heavy elements, e.g. tungsten, 
1 X.U. = lo-ii cm. 



EMISSION AND ABSORPTION 



55 



a fifth line, a 3 , has been recorded, and the complete 
spectrum of the light elements includes as many as 
seven in the a group and five in the /? group. 

In all cases the wave length of the emission lines 
was found to be independent of the chemical combina- 
tion of the element which emitted it, just as were the 
original ' characteristic ' secondary radiations. Recently, 
however, evidence has been produced of a slight varia- 
tion according to the chemical combination of the 
elements sulphur, phosphorus, and chlorine. 

Siegbahn and his workers have made a complete 
survey of the lines of the K series for the elements. 
They find that the Moseley relation holds very well 
for each of the four K components. This is graphically 
shown in Fig. 23, where it is seen that, except for 
a slight concavity upwards, the curve is very nearly 
a straight line. 

In the case of the light elements of atomic number 
of the order of, say, fifteen the K radiation is exceed- 
ingly soft and consequently easily absorbed in air. 
The production in a measurable form of such radiations 
was therefore somewhat difficult. Until recently the 
K radiation of sodium, which Siegbahn found to be of 
wave-length 11883-6 X.U., was the softest radiation of 
this series to be measured. But within the last year 
experiments have been described extending the range ; 
for example Dauvillier claims to have isolated the 
carbon K radiation by means of a vacuum spectro- 
graph. 

Previously the K radiation of the elements of atomic 
weight of the order of carbon had only been known 
indirectly by their photoelectric action on radiators 
placed in the same evacuated space in which the rays 
were produced. The conditions of excitation of these 
radiations were deduced from kinks in the curves 
showing the relation between photoelectric emission 
and applied potential. 

Even in the most extreme vacuum spectrometer work 
the size of the crystal lattice places a limit to the 
5 



56 



X-RAYS 




EMISSION AND ABSORPTION 



57 



measurable wave-length. However, as seen in the 
chapter on ' Reflection and Refraction ' other possi- 
bilities are now available for direct measurement. 

The L Series 

Fig. 24 is a tracing of a photograph by Friman of 
the L series of a few of the heavier elements which 
bears out remarks made above about the comparative 
complexity of this series. As in the case of the K 
series just considered we see a gradual shift of the lines 



Au 







r, A/» 


44 






| 








1 





























Tl 



Pb 



Fia. 24 

as a whole towards the short wave-length and of the 
spectrum as we pass to atoms of high atomic number. 
Once again the spectra are of identical form. 

The notation used for the lines of this series differs 
according to the observer, but it seems probable that 
that used by Siegbahn is on the whole preferable. The 
lines are taken in three groups ; the a group consists 
of the longer wave-lengths, and the /? and the y groups 
are successively shorter in wave-length. As in the 
case of the K series, suffixes are used within the groups 
indicating the intensity of the line ; the higher the suffix 
the less the intensity. The advantage of this system 
is that it allows the addition of any new line, which 



58 



X-RAYS 



will obviously be fainter than those already found, 
without upsetting the order of the rest. No doubt 
there is much to be said on theoretical grounds for the 
slightly different notation of Sommerfeld, but as 
Siegbahn points out the whole matter is one which 
should come under review when a more complete survey 
of theoretical and experimental conditions has been 
made. 

In the case of tungsten, which often forms the anti- 
cathode of X-ray tubes, the L spectrum has been 
thoroughly worked out, giving the following well- 
established lines in the order of decreasing wave-length : 
I, a a a lt r), t , t , p 1; p 3t p s , p 9 , p 7 , p 5> p i0> p tt ys yu yet 
Yz> Y*> Yf The I line is of much longer wave-length 
than the a 2 and seems to form a separate ' group ' : the 
same seems to be true of the rj line between the a t and 
pi in wave-length. In the case of tungsten the range 
of wave-length in the L spectrum is from 1675-05 X.U. 
(I) to 1026-47 X.U., (y 4 ), i.e. about 1-67 to 1 A.U. 

Within this comparatively large number of lines 
the range of intensity is considerable. The strongest 
line, the a, is about 120 times that of the weakest ; 
the two a lines are in the ratio of about 10 to 1. 

The Moseley diagram may be plotted for corre- 
sponding lines from the elements, i.e. the square 
root of the frequency is proportional to the atomic 
number, but the constant of proportionality is such 
that the lines of different sub-groups intersect when 
plotted. 

The L series of course is softer than the K, so that, 
as is to be expected, the lines of the series for the lighter 
atomic weight elements have not been isolated. The 
well established L lines are those from the elements 
ranging from uranium to copper. 

M and N Seeies 
The lines in the M series which are determined by the 
use of vacuum spectroscopy for atoms of atomic number 
92 (Ur) to N = 66 (Dy) have also shown a complex 



EMISSION AND ABSORPTION 



59 






structure. The determination of these lines has been 
accomplished using a mica crystal which apparently 
gives fairly sharp definition admitting a good measure- 
ment, but in this region the deviation from Bragg's 
Law 1 is large. 

The N series have been investigated by Hjalmar for 
the elements Ur, Th, Bi. The extension of these series 
to the elements of lower atomic weight is very difficult 
as absorption and the selection of suitable gratings both 
present considerable difficulty. 

General Conditions foe the Excitation oe ' Line 
Spectra ' 

In 1909 Kaye showed that in order to produce the 
characteristic radiation of an anticathode, the cathode 
rays must exceed a certain minimum velocity. This 
was investigated quantitatively by Whiddington for 
secondary rays, and by Beattie for primary radia- 
tion. 

By a magnetio spectrum method the parent cathode 
rays were sorted out into beams of one speed, which 
were allowed to fall in turn on anticathodes of different 
elements. For a given anticathode it was found that 
there was a minimum speed, below which the charac- 
teristic radiation was not excited. The value of the 
minimum velocity was found to depend on the atomic 
weight of the anticathode in a way which could be 
represented by 

v = A X 10 8 
where A is the atomic weight of the anticathode. 

More recently another interesting investigation into 
the conditions of excitation of characteristic radiation 
was made by Webster. He was able to produce a 
steady source of potential with a battery of some 
20,000 accumulators. The ionization chamber of a 
spectrometer was set in a position to receive firstly the 
Ka and secondly the K/? radiation of the anticathode 

1 See p. 21. 



60 



X-RAYS 



of the tube. In each case the voltage was gradually 
increased and the intensity of the reflected beam was 
measured. The curve representing the relation between 
the potential and the intensity showed a decided 
kink, corresponding to an increase in ionization, at 
one definite potential which was the same for both 
lines. From this he concluded that the same critical 
potential was required to excite both lines. In no case 
did the harder line appear without the softer. 

The energy of the electron in the cathode ray beam 
in the X-ray tube, to which the measured potential V 
was applied, is Ye. According to quantum theory this 
should be related to the frequency, v, of the radiation 
produced by the expression Ye = hv, where h is 
Planck's constant. It was found that this expression 
held if v is the frequency of the shorter wave-length 
component. Incidentally he was also able to show 
that the intensity of the combined a lines was always 
a constant times that of the combined /? lines. This 
and other investigations has also demonstrated that 
the intensity of individual lines varies approximately 
as the square of V. Recent experiments, however, 
appear to show that this only holds when the current 
and the applied potentials are below limiting values. 
Beyond this limit the intensity is less than that 
predicted. 

A close investigation of this nature for the L rays 
has made it clear that the complex spectrum is excited 
in its entirety provided that the potential V applied 
to the tube is that which satisfies the quantum equation 
given above in which v is the frequency of the shortest 
wave-length line emitted. There is in this case, how- 
ever, an outstanding difference from the K series, for it 
appears that part of the spectrum consisting of the long 
wave-length lines may be excited together on applying 
a smaller potential than that we have just considered. 
Indeed, there appear to be three excitation potentials 
which produce respectively three sub-groups of the L 
spectrum. That is to say when the lower potential 






EMISSION AND ABSORPTION 61 

is applied, the long wave-length lines appear. If the 
nexthigher potential is applied two sub -groups appear 
togethe? and for the highest critical pot entiaT the whob 
L spectrum appears simultaneously. The general 
properties of the K series described in the precedmg 
paragraph find a counterpart in each of the three sub- 
cttouds of the L series. 

h There are indications that in the M series the spectrum 
may be excited in five sub-groups and in all Probability 
for the N series there are seven groups each with its own 
excitation potentials. 

General Radiation 
When the radiation from an X-ray tube is reflected 
at the crystal of an ionization spectrometer the result 
is as shown in Fig. 11, i.e. it appears that the beam is 
made up of a large range of wave-lengths which form a 
continuous background upon which is superimposed the 
' lines ' of the K, L, etc., series. This corresponds to the 
case in the visible region of a white light background with 
certain bright lines enhanced. For that reason the con- 
tinuous background is sometimes called I the white 
radiation of the tube, or more often the general 
radiation. In some of the early work the areas in- 
cluded under the ionization curves such as that oi 
Fig 11 were measured, and it was found from these 
considerations that the energy in the lines was com- 
parable with that in the general radiation. However, 
if fine slits are used, it becomes evident that the energy 
in the lines is small on the whole compared with the 
total energy of the general radiation. 

From what was said in the last section it is apparent 
that the characteristic lines do not appear until the 
applied potential on the tube is at least equal to some 
minimum value ; below this potential the tube gives 
out general radiation only. In any wave-length ot 
this general radiation, however, the energy is small, 
and for ordinary purposes, unless large power is avail- 
able, measurements such as the determination of 



62 



X-RAYS 



absorption coefficients are difficult for this reason 
On the other hand, tho integrated effect of these smali 
energies in the different wave-lengths constitutes the 
greater portion of the tube output. 

If a curve is constructed showing the relation between 
the wave-length and the intensity of that wave-length 
for the constituents of the heterogeneous beam from a 
tube, one noticeable feature is that at the short wave- 
length end of the curve the radiation stops abruptly, 
i.e. the ionization becomes zero. It may also be 
observed that the ionization rises to a maximum at a 
definite wave-length and then falls gradually on the 
long wave-length side. When the applied potential is 
increased the intensity of all the components increases 
and the position of the maximum is displaced to the 
short wave-length side in the same way as the position 
of maximum intensity in black body radiation is dis- 
placed for a rise in temperature. At the same time the 
short wave-length limit is similarly displaced to the 
left. These points are all illustrated in Fig. 25, which 
are Ulrey's results obtained with a tungsten target 

The curves plotted are for ionization currents in a 
spectrometer and not for the intensity of the radiation 
Whereas these two quantities— ionization current and 
the intensity— are proportional when the range of wave- 
length used is not great, in these experiments the ranee 
is considerable and therefore the proportionality is 
not exact. Also the second order of a radiation of 
wave-length, say -25 A.U., occurs in the same position 
as the first order of a radiation of wave-length approxi- 
mately equal to -5 A.U., enhancing the ionization at 
that setting of the spectrometer; similarly all the 
longer wave-length values are increased by the higher 
orders of the short wave-length. In spite of these 
disturbing factors the curves show the general trend of 
the intensity variations. 

It might be noted in passing that the general form 
ot the white radiation 'intensity' curve has to be 






EMISSION AND ABSORPTION 



63 



taken into account when estimating the intensity of 
an emission line. For example, if in Fig. 25 the 
potential on the tube, say 50 kilo volts, were sufficient 
to produce the K series of the metal of the anti- 
cathode in addition to the general radiation, and if a 




A'nA.U.' 



Fia. 25 



line came at about -5 A.U. and actually of the order of 
six arbitrary units of intensity, it would appear to be 
of eighteen units, whereas, if the line of this intensity 
came at about -23 or -7 A.U., it would apparently be 
merely twelve units in intensity. If, however, the 
general radiation intensity curve is plotted the exact 
contribution by the monochromatic radiation alone 



64 



X-RAYS 



EMISSION AND ABSORPTION 



65 



may be readily obtained. This allowance was made 
m the determination of line intensities which were 
discussed above. 

From the point of view of the last section perhaps the 
most interesting fact to note here is that careful measure- 
ment shows the minimum wave-length A^ emitted 
at any potential V, is inversely proportional to the 
applied potential, or what comes to the same thing 
the maximum frequency Vmax is proportional V. It is 
further found that the Einstein law holds in this case 
i.e. Ve = h VaMx . This has been verified by various 
observers who have covered a range of potential from 
4,500 to 100,000 volts. 

It will be noted that this affords an accurate means 
either for the measurement of Planck's constant h 
or for the measurement of the potential applied to an 
A-ray tube. A simple modification of this principle 
has been commercially adapted in which a fluorescent 
screen spectrometer has been constructed. The limit 
on the short wave-length side is observed on the screen, 
which is calibrated both in volts and wave-length, 
bo that the device forms a direct reading instrument. 

Total Intensity op General Radiation 
The total intensity of the radiation from the anti- 
cathode of a tube, excluding the energy in the charac- 
teristic emission lines, has been found to be very nearly 
proportional to the square of the applied potential 
or since Ve = \mv* for each frequency, the energy 
of the radiation is proportional to the fourth power of 
the velocity of the parent electron. 

From the point of view of efficiency of generation 
ot A-rays, the more important consideration is that of 
the variation of the total intensity when different 
elements act as anticathode. This problem was first 
investigated by Kaye. He found that the total intensity 
ol the general radiation was approximately proportional 
to the atomic weight over a range of some twenty- 
tour elements differing greatly in atomic weight 






This proportionality while open to minor corrections in 
detail, indicates the general variation of the energy 

^iJi^altthat has been said about intensities in the 
preceding paragraph the remarks must be taken as 
giving the outline rather than the detail. For the 
determination of absolute values the individual observa- 
tions are to be corrected for various effects. From 
the nature of the experiment the radiation used covers 
a large range of wave-length, and the ionization co- 
efficients of the different wave-lengths are dtf erent 
and this is accentuated by the compulsory use of finite 
length ionization chambers. Further the reflection 
powers of crystals differ according to the ^-length 
as does the absorption in the anticathode itself and in 
the path between it and the gas in the ionization 
chamber. Experiments have been conducted by several 
experimenters who have incorporated corrections for 
the disturbing factors mentioned. As a result we 
may state that the intensity of the general radiation is 
proportional to V* for a given anticathode and that 
with reservations, the intensity is proportional to the 
atomic number of the anticathode when a constant 
potential is applied. 

Absorption 
As we saw in Chapter I the absorption of X-rays 
may be measured by the mass absorption coefficient 
all We have seen subsequently that energy may be 
taken from the beam in the ejection of photoelectrons 
and also in the process of scattermg. It seems there- 
fore suitable to sub-divide p into two parts ; x which 
is called the ' true absorption coefficient corresponding 
to the energy absorbed in the photoelectric process 
which puts the atom in a state to radiate characteristic 
radiation, and 8 which corresponds to the energy scattered 
in passing through matter. The exact apportionment 
of the absorption into the two parts is a matter of no 
little difficulty. 



66 



X-RAYS 



v™ l^K* ° f thG ^ abaor Ption as measured for 

InlhT^l kT 1S T fc the 8ame as for ver ^ na rrow oZ 

tio ITi e T a ^amount of the scattered radia 
forlTn AM et u P m the absorbing substance travel 
forward with the beam, whereas in a narrow transmit^ 
beam very ittle scattered radiation is iSude*T£ 
X-ray spectrometer, where slits are narrow we ha™ 1 
case in which the measurement of abso^tlon by L fce r 

£E? b^rTorlf^ T^ 8Ubsta - e in ^ beam 
a*2L , i / aft6r , th ° cr y stal is °™ which yields 
a true value of p so that in this case ambiguity doS 
not arise unless the screen is placed too near the 

Z;:tZ„t mh :\? hen i 8eC ° n ^y ^ationTfrom 

The tl .? akG *£? T lue ° f * a PP«« t0 ° ^all. 
,J? + dlffi l cultle8 wh ^h arise in the narrow mono 

Ml to tZ 8 ho efleCted + by the Cry8taI sP^omeTer, 
S t he W twT- at 2 m,C Wei e ht a ^orbers, are 
neceZr^ J ±1- """ *«** of these materials are 

t c r e c^ ffi owiog f the ,"i! atioa "etw^fr^rat™ 8 

tion coefficient and the wave-length is given in Fie '6 

Th wfX^ X C T- b ? h "S same a ~° 






EMISSION AND ABSORPTION 67 

of the shortest wave-length in each of the three sub- 
groups of the L spectrum which have been referred to 
above. 

Photographic Methods 

Investigation of absorption has also been made by 

photographic methods. For example, in photographs 

of emission spectra taken by de Broglie certain sharp 

edged bands of excessive blackening in the photographic 




Fio. 26 



image were eventually traced to the K absorption in 
silver and bromine which are present in the photographic 
emulsion. The sharp edge is called the 'absorption 
edge'. Calculation has shown that the frequency 
corresponding to the absorption edge is identical with 
the limit in the ionization experiments, i.e. very slightly 
greater than that of the shortest wave-length emission 
line. On that account it is clear that elements are 
specially transparent to their own radiations. 

~ e points referred to are illustrated in Fig. 27, the 



The points referred to are illustrated in Fig. 27, the 
upper half of which shows the appearance of the absorp- 
tion edge of silver as obtained in de Broglie's experi- 



68 



X-RAYS 



ments. The blackening is on the short wave-length 
side, as all the rays reflected by the crystal to this region 
are shorter than the K critical wave-length, and *on 
sequently they excite the K rays of silver in the silver 
salts contained in the emulsion, to a greater extent than 
the ' eV? 8 ^ Wave - len ^ h on the other ^de of 

If a silver absorbing sheet is placed in the path of the 
rays before they are examined, the appearance of the 
Plate is as shown in the lower half of the figure The 
long waves are absorbed in the ordinary way in the 

Direcf W Aq 3r 

Beam ^rcy Alsor/Aon /Absorfilhn 



edge 




W.L. lines 



Fig. 27 

silver sheet and fall on the plate producing correspond- 
mg blackening. Those of greater frequency than the 
critical K frequency of silver are highly absorbed in the 
interposed screen and the extra blackening which is 
to be anticipated in this region in the photographic 
emulsion due to the X-rays which get through the 
sheet 18 not sufficient to compensate for this selective 
absorption The result is a reversal of the first image 
Incidentally this provides a useful method for the 
investigation of the absorption in matter. Also un- 
desirable absorption regions are liable to appear when 
X-rays are taken at very small grazing angles from the 
surface of an anticathode. 
The three absorption edges in the L region which one 



EMISSION AND ABSORPTION 



69 






would expect from ionization experiments have also 
been found by these means for the heavier elements. 

In the case of the M absorption limit experiments 
are yet very scarce, but for uranium five edges have 
been found and further indications of a similar structure 
have been obtained in a few of the heaviest elements. 

Absorption and Wave-length 
From the collected absorption data empirical rela- 
tions have been deduced to show the connexion between 
A and p/p for absorbers of different atomic number, N. 
The expression 

p a = KN 4 A 3 + -8Ns ... (1) 
is typical of those dealing with the atomic absorption 
coefficient, fx a . In the formula s is the scattering per 
electron and K is a constant. 

When considering one absorbing substance only the 
corresponding relation is 

H/p = K'A 8 + 8/p . . . (2) 
Recent experiments have given a complete consistent 
survey for a range of wave-lengths covering both 
absorption discontinuities. Using very fine slits Richt- 
meyer, for example, has obtained values which pass 
completely through the K discontinuities and the form 
of the curve connecting absorption and wave-length is 
now well established and is illustrated in Fig. 26. 

By allowing for s/p differently, suggestions have been 
made that the power of A, in (2) above, should be 3/2, 
and subsequently the claims of other values between 
2-5 and 3 have been supported. However it now seems 
clear both from theoretical and experimental considera- 
tions that the power 3 given in (2) is highly probable. 

From theoretical considerations many absorption 
formulae have been developed. Perhaps the most 
probable in view of recent experiments, are those of 
de Broglie and Kramers. Both these yield a result 
for the true absorption, t, which is of the form sug- 
gested in the empirical relation above, i.e. KN 4 A 3 . 



70 



X-RAYS 



The value of K differs slightly in the two theories and 
both are higher than the experimental value, but 
differentiation between the theories or a final test of 
the validity of either of them over a large range of 
wave-lengths and for all absorbers is not possible until 
more precise experimental measurementss are available 
for the scattering at all wave-length and also in some 
cases more precise measurements of /z/p. 

It is possible that when s/p is thoroughly investigated, 
important deductions may be made from curves which 
express accurately the relation between r/p and X. 

An interesting addition to this subject is to be found 
in the Ph.D thesis of Johnson (Upsala, 1928). He 
found the absorption per electron (/;,) as expressed in 
the formula 

_fi A 
* ~ ~p LN' 

where A is the atomic weight, N the atomic number, 
and L the number of electrons per gramme molecule, 
and plotted ft. against X. This gives a curve similar 
in form to that of Fig. 26. 

It was found that if the values of p. between the K 
and L absorptions were multiplied by a factor Ek/El, 
the ratio of the ordinate at the K limit, the adjusted 
values gave a continuous curve when plotted against x. 

This continuous curve was obtained for all elements 
investigated. 

The value of the scattering term in (1) is too great 
in the case of short X-rays and more so in the case of 
y rays. The value of this term itself is about twice 
as great as the total observed absorption in these cases. 
This objection, however, does not apply to (2) where 
s/p is given the observed value and not that calculated 
from the classical theory (0-2), which fits a fair range for 
X-rays of medium hardness. 

The Atomic Nature of Absorption 
All mvestigations until recently apparently established 
the fact that X-ray phenomena were atomic in char- 



EMISSION AND ABSORPTION 



71 



acter. In particular early experiments had shown 
within the limits of accuracy obtainable, that the 
absorption by an atom was independent of the particular 
state of chemical combination. Within the last few 
years experiments, which were first performed by 
Beroengren, have been interpreted as showing that the 
absorption limits for sulphur, phosphorus and chlorine 
are dependent on the chemical combination of those 
elements. The position of the absorption edge has been 
shown to vary by some 10-15 XU according to the 
chemical compound used to absorb the rays. It is 
significant that these three elements are the ones which 
we have previously noted in this chapter as giving out 
emission lines of a wave-length which appears to be 
dependent on the state of chemical combination. 

Apart from these exceptions, however, we may state 
quite definitely that the atomic nature of absorption 
as a particular case of X-ray phenomena is well estab- 
lished. 



CHAPTER V 



PHOTOELECTRONS AND IONIZATION 

Introductory 

F $e e . ar, y da y« ot X-rays the electron emission 
X which is brought about by the action of X-rays 
was referred to as the 'corpuscular secondary radia- 
tion , as already mentioned in the opening chapter. 
This effect was first recorded by Perrin ; later Currie 
and bagnac showed that when metals are irradiated 
m vacuo they acquire a positive charge, due to a loss 
of negative electricity : it was shown that if a sufficiently 
great opposing electrostatic field were applied the loss 
could be entirely arrested. Finally the magnetic 
deflection of the negative particles enabled an estimate 

♦h«T m .wu , their maSS and charee > and identified 
tuem with the electron, and the process wasshown to be 

identical m character with photoelectric emission in 

the visible region Indeed, it is now referred to as 

the photoelectric effect of X-rays, and it is this process 

and some of the properties of the photoelectron which 

are to be described in the present chapter. 

Absorption and Range 
The study of the photoelectric action of X-rays is 
pursued by means of experiments carried out in vacuo. 
-bven the quickest electron is readily absorbed in air, which 
is thereby ionized; this makes measurement of charge 
etc quite unreliable without the complete evacuation 
of the containing vessel. The nature of the absorption 
is somewhat complex. Sir J. J. Thomson assumed 

72 



PHOTOELECTRONS AND IONIZATION 73 

that the photoelectrons lose energy in encounters with 
other electrons in matter, and consequently pass on with 
a reduced velocity. On these lines he was able to 
deduce the relation between the original velocity (v) 
and the final velocity (v g ) after passing through a path 
x cm. of a medium, to be of the form 

v *-v* = kx .... (1) 

where A; is a constant. 

If an electron is directed into a gas, it produces 
along its path ionization which becomes zero at a point 
which depends on the gas pressure and the original 
velocity of the electron. When the ionization ceases 
the velocity has been reduced to a negligible fraction 
of its initial value. If we measure the length of the 
path from the point of ejection to the point where it 
ceases to ionize (and for the most part where it ceases 
to have the properties by which it is recognized) we 
obtain what is called the range, r, of the particle, which 
by equation (1) is seen to be given by 

v* = kr (2) 

In the case of an electron ejected from a solid the 
velocity of emergence depends on the point of origin 
within the solid. The final velocity of emergence will 
obviously depend upon the initial velocity, the value of 
the appropriate k in the solid, and the length of the path 
in the solid, as shown in (1). If we employ a very thin 
film of the substance as source, e.g. gold leaf, we may 
eliminate this depth effect to a large extent and so 
provide a means of studying the range of photoelectrons 
in gases. Whiddington verified expressions (1) and (2) 
by experiment, and determined the values of k for 
various elements. From a study of the total range of 
the photoelectrons he was able to determine their velocity 
of ejection. He had previously shown by experiment 
that the ' K radiation ' was emitted by the anticathode 
when bombarded by electrons whose minimum velocity 
must exceed 10 8 x (Atomic weight of the anticathode), 
and was thus able to combine the two experiments and 



74 



X-RAYS 



give the following important conclusion :-The maximum 
velocity of ejection of a photoelectron produced by 

«f Jh 3 * °fi a / 1Ven wave " len g th » equal to the velocity 
oi the cathode ray originally responsible for that X-ray 
inis may be expressed in the form 

^ZKZT^ refinements , thi « means, reading left 
tonght that there is an equivalence between the < cathode 

E y i J?^ ? ( l uantum {t Produces ; reading right 

i i?,L i TTi^ l h °, ener ^ of the Photoelectfon 
a equivalent to that of the quantum which has ejected 






Experimental Methods 
The experimental methods which have been used 
to investigate the photoelectric effects of X-rays are : 

(1) Ionization method. 

(2) Stopping potential method. 

(3) Stopping magnetic field method. 

(4) Magnetic spectrum method. 

(5) Cloud condensation method. 

Li the ionization method a very thin film of the 
material to be used as source of electrons is suspended 
near the back of an ionization chamber opposite a 
thin aluminium window which is suitably strengthened 

ran*? TK ! ?? ?**? ™ thhl t0 be Varied 0Ver a ***> 

i S ! i ile 1 totaI conization at each pressure is obtained 
and plotted against the corresponding pressure as 
shown in Fig. 28(a) Now at a given pressure the total 
ionization produced in the gas is made up of two parts : 
(a) the ionization produced ' directly '2 in the gas : 
Jawi! 8 S r °P° rtional to the pressure of the gas, and 
fli u « ense conization produced in a layer near the 
film by the photoelectrons which it emits. It is clear 
that if the incident beam is constant the ionization 
due to the second cause (I e ) is independent of the 

1 See also page 81 (Richardson & Compton). 
* oee puge 78. r ' 









PHOTOELECTRONS AND IONIZATION 75 

pressure so long as all the photoelectrons are absorbed 
in the gas (that is neglecting the slight change in the 
absorption of the X-ray beam before it reaches the 




J onizafion wthouf 
end photoelecTrons 



P press i 




(b) 



Pinax. 



Fio. 28 



film). Therefore the observed ionization I is given by 

I = I. + aP 
where a is a constant, and aP is the ionization produced 
in the gas ' directly '. 

In the figure the broken line, parallel to the straight 
part of the experimental curve, and passing through 



76 



X-RAYS 



the origin, represents aP. The difference in the ordinatea 
gives I e which may be plotted as in Fig. 28(6). From 
this curve is obtained the minimum pressure at which 
all the electrons are absorbed, or, in other words, the 
pressure corresponding to maximum ionization, shown 
m the figure as Pmax. If t is the X-ray path in the 
ionization chamber, from window to radiator, and 6 is 
the temperature in degrees centigrade, the range of the 
electron at 0°C and 76 cm. pressure is 

range at N.T.P. = t Pmax 273 
76 273 + 

This method has been used in many of the earlier 
important researches. 

The method which is often called the stopping 
potential method consists in having an independent 
plate or grid, usually parallel to the emitting surface, 
and raised to a potential negative with respect to that 
surface. The negative potential is increased until 
an electrometer connected in series with the battery 
which produces the potential difference between the 
plates, shows no deflection, i.e. until the applied voltage, 
V (volts), is sufficient to stop all electrons from leaving 
the surface of the metal. In these circumstances it 
is clear that if e is the charge on the electron in electro- 
static units, and m is the mass, that 

e.V/300 = |mt) 2 
from which v may be calculated. This method is used 
in the case of very soft X-rays only. 

The magnetic stopping method is one which has 
recently been used to determine the velocity of emission 
of photoelectrons in order to calculate the wave-length 
of soft X-rays. It consists in applying a transverse 
magnetic field of such a strength that none of the 
photoelectrons reach a plate at a measured distance 
away, but are bent into circular arcs which just do not 
touch the collecting plate. The calculation of the 
velocity depends on the form of the source and the 
collector. The method has not been very much used, 












PHOTOELECTRONS AND IONIZATION 77 

but a modified form has been recently employed to 
find e/m and v for the electrons emitted by the filament 
of a thermionic valve (Hull). 1 '• • 

The magnetic spectrum method depends on the 
fact that if a charge, e, moves across a uniform magnetic 
field of strength H it is subjected to a force always 
at right angles to the field and the direction of motion, 
and of a magnitude Hey, which constrains it to move 
in a circular path. . 

The source of electrons is placed beneath a slit in 



Pump 



X Pay. 




Fig. 29 



Apparatus used In *e Production^th^MagneUc Spectrum of a source of 

a horizontal platform, on the uppei -surface ,o\ ! which 
is a photographic plate. The whole, for obvious 
reasons', is JrXed in a light-tight box, and suitable 
lead blocks prevent direct action of the X-ravs 
on the plate. When a transverse magnetic field is 
applied, the electrons which are ejected by the action 
of a beam of X-rays are bent into circular arcs, pass 
through the slit and strike the plate. Each group 
of electrons with one velocity form circles of one radius 
and meet the plate in a line as broad as the slit width^ 
Now from the measured position of the me and the 
known dimensions of the apparatus r,*fa* »f»<* 
curvature of the path may be calculated, and, since 
mv z /r = Hev, v may be obtamed, viz. : v = a.r.e/m. 
i See Appleton, The Thermionic Valve, in this series, or Physical 
Review, 1923. 



78 



X-RAYS 



Of course for highspeed electrons where v is an appreci- 
able fraction of t he velo city of light, the value of m 
is taken as m /Vl - ^ where fi has the usual signifi- 
cance ( = v/c) and m is the rest mass of the electron 
I his applies to all cases of fi rays from radioactive bodies 
f nd *° P hot °electrons eie *** 1 h 7 very short wave- 
length X-rays and by y rays. For a heterogeneous 
source there are as many images on the plate as there 
are velocities in the source and the lines constitute 
the magnetic spectrum. 

The cloud condensation method is one which gives 
direct evidence of the behaviour of electrons in the 
production of ionization, and indeed, by photographing 
the instantaneous position of individual ions it provides 
a very powerful means for the investigation of the action 
ot all ionizing agents. For example, the first published 
Pictures of C. T. R. Wilson confirmed in a striking way 
the theory that in the case of ionization in gases, the 
whole effect was due to the secondary action of the 
photoeiectrons which the X-rays liberate from the 
gas molecules. The apparatus took the form shown in 
outline m the diagram Fig. 30. AB is a glass vessel 
m the lower half of which is a plunger P, standing over 
water. Through the tube T, connexion is made to 
an evacuated reservoir, not shown in the figure. When 
the piston H is moved to the left the pressure difference 
causes a sudden depression of P and consequent expan- 
sion, and coo ing of the gas in AB. When no precautions 
have been taken, a mist is formed by this process by the 
condensation of the cooled water vapour on dust par- 
ticles m the air. The drops of water surrounding the 
dust particles form a cloud which slowly falls to the 
bottom of the vessel. When the process has been 
repeated a few times the space becomes dust free, and 
thereafter a sudden expansion and cooling of the air 
results in the supersaturation of the air, without the 
formation of mist in the space, as no centres of con- 
densation remain. It had been found previously that 
it A-rays are passed through such a dust free space, 









PHOTOELECTRONS AND IONIZATION 79 

which is cooled in the way described, water drops form 
on the ions produced. 

Wilson arranged the apparatus as shown so that ma 
dust free atmosphere a movement of the piston to the 
left causes the following sequence of events : (1) By 
sudden expansion, supersaturation of the space was 
produced when the piston H passed the tube T. (2) 




Fig. 30 



The thin thread between the weights C and D broke 
when the piston was stopped. (3) D passed between the 
spark gap 11 and caused the charged condensers F, ** 
to discharge through the X-ray tube, and so pass a beam 
of X-rays through AB. (4) After a short interval of 
time which could be adjusted by altering the distance 
between the two spark gaps, D discharged, in a similar 
way, a pair of large condensers K, K\ by reducing the 
gap 22 This caused a spark to pass in the gap S, and 



80 



X-RAYS 



so provide a strong flash of illumination on the water 
drops condensed on the ions, which were photographed 
by a camera placed at right angles to the figure opposite 
AB. 

The photographs showed that all the ionization takes 
place along definite irregular tracks, which are in fact 
an alignment of gas molecules ionized by the photo- 
electrons given out from the molecules of gas. The 
tracks in other words show the path of the photo- 
electrons. \ 

Later, the sequence of events described above was 
incorporated in an apparatus with improvement in 
detail of design, and a technique has been developed 
whereby, using a stereoscopic pair of photographs, the 
depth effects may be appreciated and some idea of the 
relative lengths of tracks may be estimated. For 
example, a short image on a single plate may be inter- 
preted as a short track or else as a long track seen end 
on : with a stereoscopic pair these may be distinguished. 
Another method consists in taking simultaneously two 
pictures at right angles to each other. This also 
enables an estimate of length and direction of the tracks 
to be made. 



Experimental Results 

The first noteworthy result obtained by the magnetic 
deflection method, by Innes (1907), showed that the 
velocity of the photoclectron was independent of the 
intensity of the X-ray beam used to eject it. This was 
found by altering the distance between the source of 
X-rays and the metal from which the electrons were 
obtained. It was shown that the intensity of the beam 
merely controlled the number of electrons given out. 
On the other hand, Beattie and Whiddington showed 
that the velocity was determined by the ' penetration ' 
of the X-rays ; the more penetrating the beam (i.e. 
the shorter the wave-length) the greater the value of v. 

The classical electro-magnetic theory could offer 
no explanation for these results. According to that 









PHOTOELECTRONS AND IONIZATION 81 

theory the ejection of the electron was brought about 
by the acceleration of the corpuscle by the electric 
vector in the wave ; and it appeared inconceivable that 
this acceleration was independent of the magmtude 
of the force. As in the case of photoelectricity pro- 
duced by light, however, the quantum theory is found 
to fit the facts in a very striking manner As we 
have already seen, this theory demands in the case 
of interchange of energy between radiation and matter^ 
that this takes place in integral multiples of a quantum 
hv Now if the quantum is all expended on the electron 
which it emits it is apparent that for one frequency in 
the incident beam there is given a fixed energy to the 
electron, which will have one definite velocity if other 
conditions are the same ; it also seems dear that for 
less intense beams there are fewer quanta available and so 
a smaller number of photoelectrons is to be expected 

Each photoelectron ejected, according to this theory, 
will take a quantum of energy from the beam, or in 
other words, the true absorption of the X-rays in matter 
should be proportional to the number of photoelectrons 
emitted. It is interesting in this c«^«on to rweall 
the results of an experiment by H. Moore (1915) who 
found by means of ionization measurements that the 
number of photoelectrons in a gas was proportional to 
the fourth power of the atomic weight of the gaseous 
atom. Taken in conjunction with the fact that the 
true absorption in matter is found to be P^P 01 * 1 ™* 1 
to the fourth power of the atomic number, N, we con- 
clude that the true absorption is proportional to tne 
number of photoelectrons emitted. 

Now considering an individual electron we see that 
the energy hv which it absorbs is used in two ways 
(a) W in dragging it to the ' surface of the atom and 
(6) (|m« 2 ) in imparting to it kinetic energy, i.e. we have 
the Einstein relation, 

hv = W +±mv* .... (3) 
an expression which Richardson and K. T. Compton 



82 



X-RAYS 



had verified for v in the visible and ultra-violet range 
Doubt was raised as to the applicability of the expres" 
sion in the X-ray region, by experiments which were 
interpreted as showing that the velocity of the electron 
was the same from whatever part of the atom it came 
However, as was pointed out by Richardson, this' 
conclusion seemed highly improbable, for it is clear that 
the value of W depends upon the level within the atom 
trom which the electron was ejected. If W E , Wj, W M 
W N . . are the energies required to extract the 
electron from the corresponding levels within the 
atom, the values of W are in a decreasing order of 
magnitude. As a result of experiments with thin 
tUms, L. Simons was the first to show in a quantitative 
manner that this is indeed the case. Simons used the 
ionization method and obtained clear indications 
that the velocity of emission was not the same The 
2 °t c ,°^P let f. answer ^ the question was provided 
by M. de Broghe, who used a form of magnetic spectrum 
apparatus very similar to that employed by Robinson 
and Rawhnson for £ particles from radioactive bodies. 
±he magnetic spectrum was obtained for the electrons 
given out by a silver radiator when irradiated bv the K 
radiations of tungsten. Fig. 31 shows in diagrammatic 
lorm the type of result obtained. 

There appears to be no doubt that the velocity has 
more than one value ; in fact, the interpretation of 
the photographs by de Broglie showed that not only 
were there present velocities given by 

\mv* = hv - W K , imv z = hv — W h 

hnt e 7w 1 thc * re( * uenc y <* the tungsten radiation, 
but that the difference between the four K lines of 
tungsten and the K level of silver (4 and 5 in the figure) 
was clear y shown. In fact, the resolution of the Ka 
and Ka 2 Imes of tungsten in this form of spectrum wJ 
Sial si?° re C ° mplete than in the ***** spectroscopic 
The values of the energies, W, obtained in these 



T 



83 



PHOTOELECTRONS AND IONIZATION 

experiments were in good agreement with those given 



by other methods. 



the 
sion 



When dealing with the case of the quicker electrons 
e form of equation used employed the relativity expres- 
mi for the kinetic energy of the electron, viz. : 

mC *(--L = =-l)=hv ai -W K {Ag) . (4) 

where v ai represents the frequency of the incident 
radiation which ejects an electron from the K level of 
a silver atom. 



l.eJ^Ag M level 
3 L levels £*L l« 



4 lines rfW 
K line Ag. 



1,1^ ^We-LAg 
L M «( 



Fig. 31 
Magnetic spectrum produced by the irradiation of silver by Tungsten K 

'"foZi) of lines corresponding to a velocity obtained from equation (4), assigned 

to the following differences of energy. 

^ and of Ag and tne three L levels of Ag. 

(2) a x and o, of „ „ ,. M level „ 

(3) of „ » .. L ... » 

(4) and (5) a, «, £, 0, of fudl levels „ 
(8) a, and o, of W „ L „ „ 
(7j o, and a, of M lovel „ 

The method has since been used extensively by 
Robinson for the accurate determination of the 
energy in the different levels of many atoms. It is 
also of interest to note that the method provides one 
of the few means for making a reliable estimate of the 
wave-length of y-rays, by the application of (4) to the 
results obtained with substances whose energy levels 
have been completely determined by X-ray methods. 

Turning our attention now to the results obtained by 
the ingenious method of C. T. R. Wilson, we find a 
considerable amount of direct information about the 
photoelectron, the recoil electron and the process 



84 



X-RAYS 



of ionization. The classical electromagnetic theory 
demands that the electron be ejected in a direction at 
right angles to the beam of X-rays, i.e. in the direction 
of the electric vector. In the photographs it is seen 
that many of the tracks start out in this direction, 
but they further show that this is not a unique direction 
of emergence. In the case of shorter wave-length 
radiation the number of electrons in the forward direc- 
tion increases. Now A. H. Compton has shown that it is 
possible with certain assumptions as to the manner of 
the interaction of the quantum and the atom, to predict 
exactly this distribution making use of a quantum 
theory. The evidence is in favour of this latter theory, 
for it shows that the increase of the number in the 
forward direction is to be anticipated as the frequency 
of the radiation is increased. 

The cloud condensation pictures show cases in which 
one point, presumably an atom in the gas, acts as the 
simultaneous starting point for more than one track, 
inis has been traced to secondary effects within the 
at °m J one . photoelectron is emitted from the K level 
and the falling in of an outer electron gives rise to the K 
radiation, which in turn ejects an L electron and so on. 
In fact this is the same kind of thing as that which 
accounts for the first few lines of the photograph 
represented in Fig. 31. 

Another feature of the photographs obtained by this 
method was the appearance of short tracks with heads 
m ?i St l y <2\ the forward direction. These tracks Wilson 
called fish tracks'. They were shown to be definitely 
short tracks and not foreshortened long ones and then- 
cause became a matter of some speculation. They are 
now associated with the recoil electron which Compton's 
theory postulates in the scattering of X-rays. 

From the equations given on pages 48 and 49, it is 
possible to deduce for the kinetic energy of the recoil 

electron, wc 2 i 



T 



\V l — ft* ~ * I ' tlie fo ^ owin g expression 



in 



PHOTOELECTRONS AND IONIZATION 

terms of 0, the angle of recoil, 

2fcv/wic 2 cos 2 



85 



Kin. En. = hv. 



\ mc 2 / 



hv 
mc' 



; COS0 l 



\ 7i to-/ I""/ ; 

from which it appears that the energy of the recoil 
electron in a direction at right angles to the incident 
beam is zero, and the maximum energy is to be expected 
when the recoil is in the direction of the incident beam, 
also for all angles, 0, the value of the energy is less than 
hv. Examination of the photographs shows that the 
lengths of the short tracks which are in the direction 
of the incident beam are longer than the rest ; it also 
seems to confirm the fact that the recoil tracks are all 
in the forward direction, but they are difficult to separate 
from the photoelectron tracks that have suffered diffusion 
by collision with molecules. 

Further support of the view that the ' fish tracks ' 
are due to the recoil electron is obtained from a count 
of the relative numbers of long tracks, due to the photo- 
electron, and the short trails of the recoil electron. 
It has been shown that the number of photoelectrons 
is proportional to the true absorption coefficient of 
the radiation, and in the same way it is to be expected 
that the number of ' fish tracks ' is proportional to the 
number of quanta scattered. Therefore it seems that 
the ratio of the numbers of the two types of tracks 
in the cloud pictures should be the same as the ratio 
of the true coefficient of absorption t to the coefficient 
of scattering s. 

With soft X-rays the fish tracks are so short that 



With soft A -rays the nsn tracKS are so snort wiau 
they are difficult to distinguish, or are not visible, and 
with -nAnptrating radiation they become difficult to 
from the long tracks. With rays 



with penetrating radiation they become 
differentiate from the long tracks. With rays of 
medium hardness, however, it is found that the relative 
number of the two types of tracks is in very fair agree- 



ment with the ratio of the two coefficients (-) 



It is 



86 



X-RAYS 






seen therefore that the cloud condensation method 
adds considerable support to the quantum theory of 
scattering. In no single case, however, was there any- 
thing to show that the recoil electron is emitted at the 
same time as a reduced quantum, as visualized in the 
Compton theory. It was indeed suggested that the 
effect might be statistical in character. This point 
was cleared up in another most ingenious experiment 
by Bothe and Geiger, who used a fi ray counter method. 
The /3 ray counter is illustrated on the left of Fig. 32, 
which is self-explanatory. When a single electron 

Xrqyt 



String 
E/ccfrorr*&r 



S 



Hl'l'l'l- 






foil 



Fia. 32 

enters at A there is set up in the space a sufficient 
ionization current (by collision) to be recorded by the 
string electrometer. 

Bothe and Geiger arranged two counters with open- 
ings (A) facing each other ; one of the openings was 
covered with a thin platinum foil, the other was free. 
A narrow beam of filtered X-rays was passed, as shown 
in the figure, through hydrogen, which acted as scat- 
terer. The recoil electron could enter the counter on 
the left, and the scattered quantum could pass through 
the foil, and if it released a photoelectron in the gas, 
could be recorded in the second counter. The time of 
arrival of the electrons was compared by photographing 






PHOTOELECTRONS AND IONIZATION 87 

the movement of the strings of the electrometers on the 
same cinematographic film, on which was superimposed 
a time scale which measured 1/1000 second and in the 
later experiments allowed an estimate of time to 1/10000 
second. All the scattered quanta did not release 
photoelectrons, but it was found when a scattered 
quantum was recorded that at the same time a recoil 
electron arrived in the other counter. This gives more 
strong support to the Compton theory of scattering, 
and from the point of view of the present chapter, 
illustrates the method which enables the measurement 
of the effects of a single electron. 






CHAPTER VI 

DIRECT REFLECTION, REFRACTION AND 
DIFFRACTION 

Early Experiments and Calculation op 
Refractive Index 

FROM the time of Roentgen many researches have 
been undertaken with the object of obtaining evi- 
dence of the reflection and refraction of X-rays. From 
our present knowledge of X-ray refraction we may con- 
sider the experiments which first seriously tackled the 
problem were those of Barkla in 1916, and of Webster 
and Clark in the same year. 

Barkla considered that if the simple dispersion formula, 
(1) below, were applicable even in a modified form for 
the high frequency X radiation, some deviation of a 
beam should be observable. He accordingly directed a 
narrow beam of X-rays through a prism of potassium 
bromide and attempted to measure the deviation. Using 
an incident beam of ' homogeneous ' bromine radiation 
estimated at -5 A.U., he came to the conclusion 
that the refractive index was between -999995 and 
1-000005. 

Webster and Clark found that the difference from 
unity in their experiments was of the order of 3 x 10" 4 
and placed their values at 10004 to 10002. 

Although the results were regarded as negative in 
character, it is clear from a consideration of the theory 
set out below that a serious attempt had been made 
to obtain conditions likely to yield results. 

The Drude-Lorentz theory of dispersion leads to the 
88 



REFLECTION, REFRACTION, DIFFRACTION 89 

following expression for the value of the refractive 
index, ft, 






u* = 1 + —2- 



n, 



2 



(1) 



Tim v* — v* 

where e and m are the charge and the mass of the 
electron, v is the frequency of the radiation incident on 
matter containing n r electrons per c.c. of natural fre- 
quency v T . The summation is carried out for all types 
r of electrons. If we consider that in the incidence of 
X-rays the frequency is large compared with the natural 
frequency of the electrons, v r 2 becomes negligible com- 
pared with v 2 and the formula reduces to 

^=1-— 2 (2) 

In this case n is the total number of electrons of all 
kinds in unit volume of the matter irradiated. 

The term — ■ r-% is small in comparison with unity, 
Timv 2 

so we may write 



/* = !-* 



ehi 



nmv 



= 1 - 



where 



<5 = 



2jttov 2 



(3) 



It will be seen that this leads to a value of /*, which is 
less than unity by the amount <5, which may be evaluated, 
by substituting the constants, for any frequency radia- 
tion. In the case of tungsten La radiation o is found 
to be 8 X 10" 6 , so that it is obvious that more refined 
methods than the ones described above are desirable 
for the detection of refraction with any certainty. 

Deviation from Bragg's Law 
The fundamental formula for the reflection at crystal 
layers, often called Bragg's Law, is 2d sin 6 = nX. In 



90 



X-RAYS 



early experiments it was shown within experimental 
error that sin 0/n was constant. 

Now C. G. Darwin as early as 1914 came to the con- 
clusion that this law is not strictly true if one allows for 
the effect of the medium (the crystal) on the radiation. 
The angle 0' given by the equation n\ Q = 2d sin 0' for 
the rath order reflection of a radiation of wave-length X Q 
in vacuo, is not the same as the observed angle 0. 

It appeared in this theoretical treatment that there 
should be refraction of the radiation as it entered the 
crystal, causing it to meet the atom layers at some 
angle differing from 0' ; also that the radiation was 
changed in wave-length to a value given by ?. /X = p. 

Extending this investigation it was found that the value 
of (sin 0)/n could be expressed in the following form : 
. /sin 0\ . , B 

where A and B are constants ( A = log g-r ). 

On the experimental side Stenstrom, in 1919, found 
for long wave-length X-rays, that the departure from 
constancy of the ratio sin 0/n was greater than his 
experimental error. The matter was more completely 
investigated by Hjalmar, working in the same laboratory. 
Using a precision spectrometer, etc., he was able to photo- 
graph and measure up to the tenth order spectrum, and 
found the variation anticipated by the theory, as is 
seen in Fig. 33, which shows one of curves expressing 
his results. It will be seen that the value of log. (sin 0/n) 
approaches a limit, for the higher values of n, which is 
the value of the Bragg equation, for, as is seen below 
in equation (5), the correcting factor becomes small as n 
increases. Also when the term B/w 2 in equation (4) 
becomes negligible, the equation reduces to 



*eM-*s) 



which is the Bragg equation. 



REFLECTION, REFRACTION, DIFFRACTION 91 

Duane and Patterson also found this deviation from 
the law in the case of rays from a tungsten target tube 
reflected from calcite. 

A theoretical expression was obtained to relate the 




I Z 3 4 5 6 r 8 9 'O 



Fia. 33 



values of the wave-length as calculated from the Bragg 
equation with the true value, 



*-*[»-*£] ' • 



(5) 



where p is the density of the crystal, n the order and 
d the lattice constant. 






92 



X-RAYS 



If we write this in the Bragg equation, we have 
HA„ = 2rf[l-^]sin0, 

from which it appears that we may regard the bracket 
expression as a correcting factor to the value of d for 
any order reflection. The use of such a corrected lattice 
constant is not justified in the case of many crystals. 
In general it is used in those ' precision ' measure- 
ments of wave-length in which well worked out crystals 
are employed, but from the point of view of the deter- 
mination of the refractive index it is of interest to note 
that if the apparent wave-length is found for two orders, 
say n = 1 and n = 2, we may calculate /x ; for example, 
Compton l calculates the value of <5 from two apparent 
values of the wave-length X x and A 2 , as obtained by the 
experiment of Duane and Patterson, by substitution in 
the expression 



d = 



X t -X t 



n, 



sin 2 0, 






A 2 »,* - n x * 

The values obtained for the refractive index of La, 
L/3, Ly, etc., of tungsten in calcite were in good agree- 
ment with the values calculated directly from (3). 

Reflection 

Since the refractive index of X-rays is less than unity, 
it is apparent that the case of X-rays falling on a polished 
mirror is analogous to the case of light going from an 
optically dense to a less dense medium : there should 
be a critical angle, and total ' internal ' reflection should 
take place when the angle of incidence exceeds this 
angle. 

If 8 e is the critical glancing angle we have cos C = (a 



since ii = sin f ^ — e ysin p. 



1 Compton, X-raya and EJectrone, p. 213. 






REFLECTION, REFRACTION, DIFFRACTION 93 

From the estimated value of fj it is clear that t is 
small, and we may therefore write 

cos 6. = 1 - 0.72 = H 
;. sin 0. = 0. = V2(l - p) = V2d 
Combining with (3) we have 

*„YJL.y£ .... (6) 

\7imJ c 
This matter was first investigated by A. H. Compton, 
who used apparatus similar to that shown m Fig. te. 
The mirror was mounted on a spectrometer table and 




Fio. 34 



by means of slits S x and S, the incident beam was 
limited to about 2 minutes of arc. The position of the 
slit S was adjustable by a micrometer screw arrange- 
ment, and when necessary the reflected rays which 
passed through it could be examined in a spectrometer 
the crystal of which is shown in the diagram. He found 
that reflection occurred at all angles less than the critical 
angle, and when the reflected beam was examined in 
the spectrometer it was shown that the intensity of any 
wave-length in that beam fell off very rapidly when 
the calculated critical angle was exceeded. It was 
also shown, as predicted in (6), that 0. was proportional 
to A. 



94 



X-RAYS 
Refraction by Prisms 



Pfafk 



Piatt 




- Reflected*. 



I RefracTed 
■*■ Direct— * 






Fig. 35 



«,,£"E! y JL m 0n ^IP^ 8 ? (L to R) nnd K ,ve rl8e to a reflected ray, and to refracted 
an y nL'?„ P08lti ? , l1 Wh ^ h . depen . d . on . the ™ve-lcngths in Oh incident beam The 
appearance of the photographic plate is seen on the right. 

some refracted. Clear photographic images were ob- 
tained of the rays shown in the diagram and the refracted 
rays of iron Ko and Kfi, also copper Ka and Kfi were 
definitely resolved, as is indicated in the right-hand 
part of the diagram, the deviations being of the order 
anticipated from their wave-lengths. It is to be noticed 
in the diagram that the rays fell on the prism at small 
angles of grazing incidence. 

Reflection Grating 
Another very interesting application of the reflection 
of X-rays was to be had in an experiment carried out 



Following these reflection experiments, Siegbahn and 
his collaborators have given a description of experiments 
in which direct refraction in prisms has been obtained. 
The refracted ray is bent away from the base of the 
prism as would be expected from our knowledge of /x. 
Fig. 35 shows the scheme of those experiments which 
combine most of the points described above. A com- 
posite beam of X-rays was directed on the prism from 
left to right. Some of the radiation was reflected and 









REFLECTION, REFRACTION, DIFFRACTION 95 

by Compton and Doan. A reflection grating ruled with 
500 lines per centimetre was set up, as in the corre- 
sponding optical experiment, with a photographic plate 
arranged to receive any diffraction pattern obtained, and 
a narrow homogeneous beam of X-rays, -707 A.U. in 
wave-length, was directed on to it at an angle within 
the critical glancing angle. The photographs taken 
showed, in addition to the direct and reflected beams, 
as many as three diffracted lines. Applying the ordinary 
grating formula 

nX = d(sin 6 r* sin i) 
where d is the grating constant, calculation showed that 
A was of the same order 







/ 1 / Z3 

Diffict , 
Rejected 



as the value calculated 
from the spectrometer 
data. 

Whereas the early re- 
sults could not claim any 
high degree of precision, 
their fairly close agree- 
ment with the spectro- 
meter determinations 
confirmed the method of 
crystal analysis and the 
deduction of the lattice 
constant. 

Similar diffraction ex- 
periments have since been performed by Thibaud and 
by Hunt, and quite recently Osgood has made use of 
concave gratings and has extended the range of useful- 
ness of this method of measurement to wave-lengths 
of the order of 200 A.U. At the time of writing the 
X-ray vacuum spectrometer has been placed on the 
market, and the use of the diffraction grating has been 
so much improved that this method is to be regarded 
as one of the most direct, and one which will eventu- 
ally be largely used for precision measurements. 



Fig. 36 

A typical result obtained with a re- 
flection grating. In addition to the 
direct reflected ray there are three orders 
on the one side and one on the other. 
D is the direct ray which has passed 
through the diilractlon grating. 



BIBLIOGRAPHY 



The following books are recommended to those who wish 
to read in detail the matter outlined in the preceding pages. 



Title. 


Author. 


Published by 


Matter dealt 
with. 


Rdntgen Raya . . 


Collected 


American 


Historical 




papers 


Book Co. 


papers 


X-raya .... 


Kaye 


Longmans 


Chaps. I and 






Green 


n of this 
book 


X-raya and Crystal 
Structure 


W. H. and 

W. L. 

Bragg 
M. Siegbahn 


Bell 


Chap. II of 
this book 


The Spectroacopy of 
X-raya (1925) 


Oxford Univ. 
Press 


Chaps. II & 
IV of this 
book 


X-raya and Electrona 


A. H. Comp- 
ton 


McMillan 


Chaps, m & 
V of this 
book 


La Technique dea 


A. Dauvillier 


Soc. Fr. de 




Rayona X 




Physique 
A. Blan- 
chard, 
Paris 




Lea Rayona X . 


M. de Broglie 


Do. 


Chaps. IV & 

v 


X-raya .... 


Do. 


English 

Translation 
Methuen 




Atomic Structure and 


Sommerfeld 


Methuen 


Chaps. II. 


Spectral Lines, 






Ill* IV 


Chap. Ill 









97 



INDEX 



Absorption, 65 

coefficients, 7 

true, 65 

frequency, critical, 62 

and wave-length, 69 

atomic nature of, 70 
Andrade, 39 
Angstrom Unit (A.U.), 37 
Appleton, 77 
Atomic number, 32 
Axial ratios, 25 

Barkla, 10, 14, 15, 88 

and Sadler, 44, 45 
Beattie, 59, 80 
Bohr, 33 

theory of, 34 
Bothe and Geioer, 86 
Bragg, W. H. and W. L., 18 
Bragg '8 law, 21 

deviation from, 89 

7-rays, 

measurement of wave-length 

of, 83 

scattering of, 52 
Change of wave-length on 

scattering, 46 
Characteristic radiations, 10 
Cloud condensation method, 78 
Compton, A. H., 38, 43 

theory of scattering, 47-52 

reflection of X-rays, 92 

and Doan, 95 

and Hazelnow, 15 
Coolidge tube, 4 
Counter, ray, 86 
Critical absorption frequency, 66 



Crystal structure, 25 
Cube centred crystal, 17 
Cubic crystals, 17 
CtJRRIE, 74 

Darwin, 90 

DATTVIIiLIER, 55 

de Broglie, M., 66, 69 
Deviation from Bragg 's law, 89 
Diffraction grating, 94 
Drude-Lorentz dispersion for- 
mula, 90 
Doane and Patterson, 91, 92 

Electronic radiation, 10 
Electron, recoil, 52 
Emission linos, 53 
Excitation of characteristic 
radiation, 12 

Face centred cube, 17 

' Fish ' tracks, 84 

Focal spot, 2 

Frequency, critical absorption, 

66 
Friedrich and Knipping, 16 
Friman, 57 

Gas tube, 4 
General radiation, 61 
total intensity of, 64 

Hard X-rays, 2 
Hafnium, 33 

Heating of anticathode, 3 
Hewlett, 42 
Hjalmar, 59, 90 



99 



100 X-RAYS 



Homogeneous radiation, 9 
Hunt, 95 

Indices of crystal, 26 

Innes, 80 

Intensity, 5 

Inverse square law, 6 

Ionization method 

Measurement of intensity, 5 
velocity of photoolectrons, 74 

J radiation, 11, 46 
phenomenon, 47 
Johnson, 70 

Kseries, 9, 11, 53 
Kaye, 12 

L series, 9, 10, 57 
Lattice constant, 20 

Determination of, 26 
Laue, 16 

LlLIENFELD, 4 

Linear absorption coefficient, 7 

M series, 58 

Mass absorption coefficient, 9 
Magnetic spectrum, 59, 77 
Magnetic stopping method, 76 
Maximum frequency, 62 
Minim um wave-length, 62 
Moore, H., 81 
Moseley's Law, 32 

N series, 58 

Osgood, 95 

Perbin, 74 

Photoelectric effect, 74 
Photographic method, 67 
Planck's constant, 47 
Polarization of the primary 

rays, 13 

secondary radiation, 15 

Recoil electrons, 49, 52 
Reflection at crystals, 19 

at grating, 95 

at mirrors, 92 



Refraction by prisms, 88, 94 
Relativity correction for mass. 

48 
Richardson, O. W., 82 

and Compton, K. T., 81 
Robinson and Rawlingsow. 

82 
Ross, 51 
Rutherford, 32, 39 

Sadler, 10 

Secondary radiation, 9 
Scattered radiation, 10 

distribution of, 41 

excess for long wave-length, 
45 
Scattering 

coefficient of, 44 

classical theory of, 40 

Compton's theory of, 47 
Schroedinger, 52 
Seeobahn, 37, 65, 67 

refraction at prism, 94 
Simons, L., 82 
Soft X-rays, 3 
Softener, 3 
Sputtering, 3 
Sommebfeld's notation for X- 

ray spectra, 68 
Space lattice, 17 
Spectrometer, 21 

self-recording, 38 

vacuum, 39, 55 
StenstrOm, 90 
Stopping potential, 76 
magnetic field, 76 

Thibauxd, 95 
Thomson, J. J., 13 

theory of scattering, 40 
determination of v for elec- 
trons, 72 
Total intensity of general radi- 
ation, 64 
True coefficient of absorption, 
65 

Ulrey, 62 



INDEX 

Vacuum spectrometer, 39, 55 



101 



Wave-length 

change of, 46 

determination of, 26 

technique, 37 

of soft X-rays, 53 
Wave nature of X-rays, 16 



Webster, 69 

and Clark, 88 
Whiddington, 59, 80 
White radiation, 61 
Wilson, C. T. R., 78, 83 
Woo, 51 

X-unit, 37 






Printed in Great Britain 

by Butler & Tanner Ltd., 

Frame and London 



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